Some of the math books recommended by Lindahua

Some of the math books recommended by Lindahua

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 1.  linear algebra   (Linear Algebra): I think the domestic college students will learn this course, but not every teacher can carry out its essentials. This discipline is necessary for learning, and it is essential for its thorough mastery. I studied this course when I was in the first year of Hkust, and after I arrived in Hong Kong, I read the linear algebra again, reading Introduction to Linear Algebra (3rd Ed).   by Gilbert Strang. This book is the textbook used by MIT's Linear Algebra class, and is also a classic textbook chosen by many other universities. Its difficulty is moderate, the explanation is clear, it is important to discuss many core concepts more thoroughly. I personally think that learning linear algebra, the most important thing is not to be proficient in matrix operations and the equation of the method-these in the actual work of Matlab can do, the key is to deeply understand a few basic and important concepts: subspace (subspace), orthogonal (orthogonality), Eigenvalues and eigenvectors (eigenvalues and eigenvectors), and linear transformations (Linear transform). From my point of view, the quality of an online textbook is whether it can give sufficient attention to these fundamental concepts, and whether it can be made clear about their links. Strang's book is doing very well in this respect. Moreover, this book has a unique advantage. The author of the book Teaches linear algebra classes (18.06) in MIT for a long time, and the course video is available on the MIT Open Courseware website. Friends who have time can watch the video of the teacher's lectures while studying or reviewing the textbook. http://ocw.mit.edu/ocwweb/mathematics/18-06spring-2005/coursehome/index.htm 2.  Probability and statistics   ( Probability and Statistics): There are many introductory textbooks for probability theory and statistics, and I don't have any special recommendations at the moment. What I want to introduce here is a basic textbook on multivariate statistics: Applied multivariate statistical analysis  (5th Ed.)   by Richard A. Johnson and Dean W. Wichern This book was used for learning when I was in contact with vector statistics, and the basis for my study in Hong Kong was to lay it down. Some students in the lab also borrowed this book to learn vector statistics. This book has no particular pursuit of mathematical depth., but to speak the main basic concepts in an easy-to-understand way, it is very comfortable to read, and the content is very practical. For linear regression, factor analysis, principal component Analysis (PCA), and canonical component Analysis (CCA) These learning The basic methods in this paper have also been discussed preliminarily. After that, Bayesian statistics and graphical models can be further studied. An ideal book is introduction to graphical Models (draft version) .  by M. Jordan and C. Bishop. I don't know if this book has been published (do not and learning in G Raphical models confused, that is a collection of essays, not suitable for beginners). This book from the basic Bayesian statistical model has been deep into the complex statistical network estimation and inference, in simple, statistical learning many important aspects of this book have a clear exposition and detailed explanation. Inside MIT you can access, as well as the outside, it seems to have an electronic version.  3.  Analysis  : I think everyone has studied calculus or mathematical analysis in college, and the depth and breadth vary with each school. This field is the basis of many disciplines, the recommended textbook is principles of mathematical analysis, by Walter Rudin a bit old, but absolutely classic, thorough. Disadvantage is more difficult-this is Rudin's book's consistent style, suitable for a certain basis to look back. In analyzing this direction, the next step is functional analysis (functional). Introductory functional analysis with applications, by Erwin Kreyszig. Suitable as the basic teaching material of the functional, easy to cut into without losing all-round. I especially like it. Special attention is paid to spectral theory and operator theory, which is particularly important for the study of learning. Rudin also has a book on functional analysis, the book may be more profound in mathematics, but not easy to get started, the content and learning the relevance of the book. In the analysis of this direction, there is an important subject is the measurement theory (Measure theory), but I have read the book is not yet felt particularly worthy of introduction.  4.&nBSP; topology   (topology): The basic Topology books I have read each have their own characteristics, but in general, I am Most Admired: Topology (2nd Ed.)   by James Munkres This book is the culmination of Professor Munkres's long-term coaching of MIT's topological classes. There is a comprehensive introduction to general topology and a modest discussion of algebraic topologies (algebraic topology). This book does not need special mathematical knowledge can begin to learn, from the basic concept of set theory (many books disdain to say this) to Nagata-smirnov theorem and Tychonoff theorem and other deep theorems (many books avoid this) are covered. The narrative way is very strong, for many theorems, in addition to the proof process and guide you to think behind the principle of the context, a lot of amazing highlights-I often read to forget hunger, do not want to addictive. Many of the exercises are quite standard.  5.  manifold theory   (manifold theory): For the topology and analysis of certain certainty, you can begin to learn the manifold theory, otherwise the study can only flow in the superficial. The book I am using is introduction to Smooth manifolds.  by John M. Lee although the title has introduction the word, but in fact the book involved in very deep, in addition to teaching the basic manifold,  tangent Space, bundle, sub-manifold and so on, also explored such as outline theory (Category theory), the DRAM on the homology (De Rham cohomology) and integral manifolds, and some more advanced topics. There are quite a lot of discussions about Lie groups and Li algebra. It is popular and rigorous, but it needs to be familiar with certain marking methods. Although Lie groups are based on the concept of smooth manifolds, it is possible to learn the Lie groups and Lie algebras directly from the matrix-a method that may be more practical for those who are in urgent need of solving problems with Lie groups. Moreover, it is beneficial to deepen understanding of a problem from different perspectives. The following book is a model of this direction: Lie Groups, Lie algebras, and Representations:an elementary introduction.  by Brian C. Hall This book is cut from the matrix from the beginning into the concept of matrix Lie groups from algebraic rather than geometrical angles. And the exponential mapping is established by defining the operation, and the Lie algebra is introduced in this way.This method is more acceptable than the traditional way of defining Lie algebra through the "Left Invariant vector field (left-invariant vector field)", and it is easier to reveal the meaning of the Lie algebra. Finally, there is a special discussion linking this new definition to the traditional way. ———————————————————————————— Whether it is the study of vision, learning or other disciplines, mathematics is ultimately the foundation. Learning maths Well is the cornerstone of good research. The key to learning maths is your own efforts, but choosing a good book is very useful. Different people have different knowledge backgrounds, thinking habits and research direction, so the choice of the book also varies with each person, just to suit themselves, do not have to insist on the same. The above book is only from my personal point of view, my reading experience is very limited, there are likely to be better than their books (may also tell me, first say thank you). The establishment of algebraic structure in  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%learning learning is a field integrating multiple mathematics into one. Speaking of mathematical disciplines related to this, we may quickly associate linear algebra with statistical models based on vector space--in fact, the mainstream papers are really based on them to a large extent. R^n (n-dimensional real vector space)   is the space we see most in paper, it's really important and practical, but it's not enough to just rely on it to describe our world. In fact, mathematicians have provided us with a much richer tool. Space, which is an interesting noun that almost appears in the basic definition of all mathematical branches. To sum up, the so-called space refers to a set and a certain mathematical structure defined above. The definition or axiom of this mathematical structure becomes the basis of this branch of mathematics, and it all unfolds. Let's start with the space--r^n  that we know best. When you use this space, other mathematical structures, including metric structures and inner product structures, are used in addition to linear operations.                      first, it is a topological space (topological spaces). And from a topological point of view, with very good properties:normal  (implying Hausdorff and Regular),  locallY compact, paracompact, with countable basis, simply connected (implying connected and path connected), & nbsp metrizable.                       second, it is a metric space (Metric spaces). We can calculate the distance of any two points above.                      Thirdly, it is a finite dimensional vector space (finite dimensional spaces). Therefore, we can perform algebraic operations (addition and multiplication) on the elements inside, and we can also give it a finite set of bases, so that each element can be expressed in finite dimensional coordinates.                      the analysis system can be established based on the metric structure and the linear operation structure. We can differentiate, integrate, establish and solve differential equations, and perform Fourier transform and wavelet analysis on continuous functions.                      V, it is a Hilbert space (i.e., a complete inner product space) (Hilbert spaces, full inner product Space). It has a very handy calculation of the inner product (inner product) structure--The measurement structure of this space is actually induced from the inner product structure. More importantly, it is complete--representing any Cauchy sequence (Cauchy sequence) has a veryLimit--many people have actually used this feature intentionally, but habitually take it for granted.                      VI, the operator space above the linear map is still a finite dimension-a very important benefit is that all linear mappings can be represented by a matrix. In particular, because it is a finite dimensional complete space, its functional space and itself is isomorphic, but also r^n. Thus, their spectral structure can be obtained by the eigenvalues and eigenvectors of the matrix.                      seventh, it is a measure of space--the size of a subset (area/volume) can be calculated. That's why we could build a probability distribution (distribution) on top of it--the basis of most of the continuous statistical models we've been exposed to. We can see that this is a very perfect space for our application to provide all the convenience mathematically, on top of which we can take for granted that it has all the good qualities we want without special proof; we can directly use its various computing structures without having to build from scratch , and many of the concepts that are not the same have become equivalent here, so we no longer need to discern their differences. In this context, the main work of learning is divided into two large categories:1.     establish a form of expression, let it in the above discussed r^n space inside. 2.     obtains the finite dimension vector expression, establishes each kind of algebraic algorithm or the statistical model carries on the analysis and the processing. Only the first category is discussed here. First of all, some of the more widely used methods:1.     directly based on the original data to establish the expression. The ultimate goal we care about is a real-world object: A picture, a voice, an article, a transaction record, and so on. Most of these things do not have a numerical vector attached to them. In order to construct a vector expression, we can put the values recorded in the sensor, or some other way of collecting the numerical data in a certain order listed in the form of a vector. If there are n numbers, it is assumed that they are inside the r^n. However, this is a little mathematically problematic, in most cases, depending on the number ofAccording to the physical principle, the range of these vectors does not fill the entire space. The pixel value of the image is generally positive and in a bounded closed set. The problem is that the likely result of a linear operation on them would spill over into the normal range--in most paper, it might simply be done in some heuristics way, or not at all, and seldom seen in much of a mathematical way-- But if you can solve the actual problem, it is also understandable, after all, not all the work needs to be as pure as the pursuit of rigorous mathematics. 2.     quantization (quantization). This is a widely used way of dealing with continuous signals. Just used to, generally do not mention the name only. For example, a spatial signal (image in vision) or a time signal, their domain values are infinitely infinite (uncountably infinite), not expressed as finite dimensional vectors, even if expressed as an infinite sequence is not possible. In this case, generally in a finite field, according to a certain number of points at a certain distance to represent the point around it, thus forming a finite dimension of the expression. This is the quantization of the signal in the time domain or the airspace. This inevitably leads to the loss of information. However, due to the high correlation of signal in small neighborhood, the degree of information loss is often not significant. And, theoretically, this is equivalent to a low pass rate in the frequency domain. For a continuous signal with limited energy, it is not possible to maintain enough strength in an infinitely high frequency domain, as long as the sampling density is sufficient, the lost thing can be arbitrarily less. In addition to expressing signals, the expression of geometric forms is often quantified, such as curve and surface. 3.     finding a finite number to adequately express an object may not be the most difficult. However, it is not necessary to build a mathematical structure on top of it. In general, we want to deal with it, first we need a topological structure to describe how the points in the space are linked together. It is often difficult and not practical to build topological structures directly in mathematics. Therefore, most of the work is done by first establishing a measurement structure. A metric space whose measurements naturally induce a topological structure-but in many cases we seem to ignore it. The simplest case is to use the Euclidean distance (Euclidean distance) expressed by the original vector as the metric. However, due to the different characteristics of the original expression values, the effect is generally not particularly good, may not be able to effectively express the actual object similarity (or non-similarity). As a result, a lot of work will have to be built on this basis to measure two times. The way is varied, one is to seek a mapping, to transform the elements of the original space into a new space, where Euclidean distance becomes more appropriate. This mapping plays a role that includes information about theTo filter, integrate, strengthen or suppress certain parts. This is what most of the articles about feature selection,feature extraction, or subspace learning, are going to do. Another way is to directly adjust the distance calculation method (some articles are called metric Learning). These two ways may not be different. If the mapping is a single shot, it is equivalent to establishing a different metric in the original space. Conversely, a measure established by changing the distance calculation corresponds to a certain mapping under certain conditions. 4.     It may be noted that the above-mentioned measure-building method, such as Euclidean distance, requires an algebraic operation of the element. For the normal vector space, the linear operation is endowed naturally, we do not need to establish specifically, so we can directly measure the construction-this is the basis for most of the work. However, some things whose original expression is not a n-tuple, it may be a set, a graph, or something special object. How to build an algebraic operation? One way is to build directly. is to define their own addition and multiplication for these things. This is often not so straightforward (the easy-to-build linear operations structure is already well established and widely used) and may involve deep mathematical knowledge and a deep understanding of the problem itself and mathematical insight. However, once a new algebraic structure is established, other mathematical structures, including topology, measurement, analysis, and inner product structures can be naturally induced, and we have the basis for various mathematical operations and operations on this object space. The addition and multiplication looks simple, but if we build these two things for the space that we do not know how to do the addition and multiplication, the theoretical contribution is very great. (A small problem: we often use a variety of graphical model, but each of these model is formulate respectively, and then deduced the estimation and evaluation step method.) Is it possible to establish some kind of algebraic structure for the space of graphical model or a specific subset of it? (not necessarily linear spaces, such as groups, rings, broad groups,  etc) so that they are unified in the algebraic sense, and the corresponding estimation or evaluation can also be used over the algebraic operation derive. This is not the scope of my research, but also beyond my current ability and knowledge level, but I believe it in the theoretical significance, to leave a vision of the problem. In fact, there really is a branch in mathematics called  algebraic statistics  that might be exploring a similar problem, but I'm on it now.The solution is very limited. 5.     back to our point, in addition to directly establishing the operation definition, another way is to embed (embedding) into a vector space, thus inheriting its operational structure for me. Of course, this embedding is not a mess, it needs to maintain some of the original relationship of these objects. The most common is the margin embedding (isometric embedding), we first set up a metric structure (bypassing the vector expression, directly to the distance of two objects by some method to calculate), and then the space is embedded into the target space, usually a finite dimensional vector space, it is required to keep the measurement unchanged. "Embedding" is a kind of mathematical application of a wide range of means, its main goal is to embed into a well-structured space, so that the use of some of its structure or computing system. In topology, embedding into metric space is an important means of establishing a measure between a certain extension outsmarted. And here, we are in the case of having a measure, by embedding the structure to get the linear operation. In addition to this, there is a relatively hot in the previous years manifold embedding, this is by maintaining the local structure of the embedding, get the global structure, will be mentioned later. The next important algebraic structure of 6.     is the inner product (inner product) structure. Once the inner product structure is established, a good metric is directly induced, which is the norm (norm), and then the topological structure is induced. In general, the inner product needs to be built on the basis of a linear space, otherwise even a two-dollar operation is the inner product can not be verified. However, kernel theory points out that for a space, as long as the definition of a positive definite nucleus (positive kernel)-a two-dollar operation that conforms to the positive definite condition, there must be a Hilbert space, the inner product operation is equivalent to the nuclear operation. The significance of this conclusion is that we can bypass the linear space and induce a linear space (called the regenerated kernel Hilbert space  reproducing Kernel Hilbert spaces) by first defining the Kernel method. So we naturally get the metric structure and the linear operation structure we need. This is the foundation of kernel theory. In many textbooks, the two-dimensional space is transformed into three dimensions with two cores, and then the kernel is used for ascending dimension. I have always thought that this is misleading to a certain extent. In fact, the most important significance of kernel is the establishment (or transformation) of the inner product, which induces a more favorable expression of the measurement and computation structure. For a problem, it is more important to choose a problem-kernel than to focus on "ascending dimension". Kernel is considered as an important means of nonlinearity, and is used to deal with non-Gaussian data distribution.That makes sense. In this sense, the structure and the structure of the original space are not linearly correlated with the inner product space of the nonlinear kernel transformation. However, we should also understand that its ultimate goal is to return to the linear space, the new inner product space is still a linear space, once it is established, the subsequent operations are linear, therefore, the use of kernel is to seek a new linear space, making the linear operation more reasonable-- In the end, the transformation of nonlinearity is still a service to the linear operation. It is worth mentioning that Kernelization is essentially an embedding process: to establish an inner product structure for a space and embed it in a high dimensional linear space in a way that keeps the inner product structure intact, thus inheriting its linear computing system. 7.     above is the process of building an algebraic structure from a global approach, but that must be based on a global structure (whether it is a pre-defined operation, a measure or an inner product, must be applied to the whole space). However, the global structure does not necessarily exist or fit, while local structures are often much simpler and more convenient. Here is a strategy that is local and global-this is the idea of manifolds (manifold), which is rooted in topology. From a topological point of view, manifolds are a very good topological space: conforming to the Hausdorff separation axiom (any different two points can be separated by disjoint neighbors), conforms to the second axiom (with a number of topological bases), and, more importantly, locally with the embryo in r^n. Therefore, a regular (Regular) manifold has the most favorable topological characteristics. And the local embryo in the r^n, representing at least in the local can inherit the various structures of r^n, such as linear operations and internal product, so as to establish an analytical system. In fact, the topological manifold inherits these structures, which is the focus of modern manifold theory research. By inheriting the manifold of the analytic system, the micro-manifold (differential manifold) is formed, which is the core of modern differential geometry. And the tangent space on each point of the micro-manifold (Tangent spaces), the system of linear operation is obtained. While further inheriting the manifold of the local inner product structure, the Riemannian manifold (Riemann manifold) is formed, and the global measurement system of the manifold-geodesic distance (geodesics) is obtained by extending the local measurement. Further, when the topology of the popular itself and the linear structure on the tangent space are connected--a cluster of topological associated linear space-vector bundles (vectors bundle) is obtained. Although manifold theory as the core of modern geometry, is a broad and profound field, but its application in the learning is very narrow. In fact, for manifold, many do LeaRning's friends first reacted to Isomap, LLE, Eigenmap and other algorithms. These belong to embedding. Of course, this is indeed an important aspect of manifold theory. Strictly speaking, this requirement is a differential-to-embryo mapping from the original space to its image, so that the embedded space has the same analytic structure on the part, and it also obtains various benefits-global linear operation and measurement. However, this concept has been considerably relaxed in the application of learning-the differential and embryo cannot be fully guaranteed, and the entire analytical structure cannot be fully maintained. People are more concerned with maintaining some aspect of the local structure--but this is understandable in terms of the tradeoffs in practical applications. It turns out that these algorithms work well when the data in the original space is dense enough. The real problem with learning is that it has been overused in sparse spaces (Sparse space), in fact, in the high-dimensional space in the thousands of or even hundreds of thousands of points, even if the most adjacent points are difficult to call local, the scope of the local and global scope actually has no fundamental difference, Even if the concept of the local can not stand the feet of the time, the back based on the work of all of them do not have much significance. In fact, the sparse space has its own laws and laws, through the local formation of the global manifold idea is not inherently suitable for this. Although Manifold is a very beautiful theory, the beautiful theory needs to be used-it should be used to solve the low-dimensional space with dense data distribution. As for some paper, the use of manifold methods in high-dimensional spaces (such as human faces) is not necessarily due to the role of "manifold" itself, but possibly other factors. The 8.     manifold plays an important role in the practical application of two aspects: one is the study of the nature of geometry (we will not talk about this), and it is the combination of algebraic structure of the formation of Lie groups (Lie group) and Lie algebra (Lie Algebra). When we study the object is the transformation itself, they constitute the space is special, such as all the subspace projection formed the Grassmann manifold, all the reversible linear operators, or affine operators, but also formed their own manifolds. The most important operation for them is the combination of transformations, not the addition number multiplication, so that the more appropriate algebraic structure defined above should be a group and not a linear space. And the combination of group and micro-manifold--Lie groups become their most suitable description system--and their tangent space constitutes a strengthened linear space: Lie algebra, which is used to describe its local variation characteristics. The relationship between Li algebra and Lie groups is very beautiful. It transforms the micro-change into the algebraic operation of linear space, making it possible to transplant traditional linear space-based models and algorithms into Li space. and the matrix in the Lie algebra is even more reflective of the change than the matrix of the transformation itself.Characteristics of the switch. The spectral structure of the Lie algebra matrix of geometric transformations can be easily used to analyze the geometric characteristics of the transformation. Finally, to summarize the strategy of embedding this widely used, isometry, kernel and manifold embedding in learning are in this category, respectively, by preserving the measurement structure of the original space, The inner product structure and the local structure are used to obtain the embedding of the target (usually the vector space), so as to obtain the global coordinate expression, the linear operation and the measure, which can be applied by various linear algorithms and models. There are also areas that deserve our attention as we gain this range of benefits. First, embedding is just a mathematical method, and it cannot replace the research and analysis of the problem itself. An inappropriate primitive structure or embedding strategy, often even counterproductive-such as the embedding of sparse spaces, or the selection of inappropriate kernel. In addition, embedding is suitable for analysis and may not be suitable for reconstruction or synthesis. This is because embedding is a single shot (injection), the target space is not every point and the original space can effectively correspond. Embedded operations often break the limits imposed by the original space. For example, two elements, even if they are mapped from the original space, and they may not have the original image, then can not directly back to the original space. Of course it is possible to consider finding a point in the original space with its recent mapping, but the validity of this in practice is questionable. The mathematical world related to learning is very extensive, and as I study and study deeply, I find that there are some structures and methods which are suitable for the problem in some branches of mathematics which I usually don't pay attention to. For example, Broad group (groupoid) and broad algebra (algebroid) can overcome some of the difficulties of Lie groups and Lie algebras in the process of representing continuous transformations-these difficulties have plagued me for a long time. Solving problems and building mathematical models are mutually reinforcing, on the one hand, a clear problem will give us a clear goal to find the right mathematical structure, on the other hand, the understanding of the mathematical structure of the problem of guiding the solution also has an important role. For solving a problem, the choice of mathematical tools is the most important, not the advanced, but if the existing mathematical methods in difficult times, the search for a higher level of mathematical help, often can steady. Many of the problems mathematicians have worked on for a long time are not all theoretical games, and their solutions often contain what we need and may lead to more problems-but we need time to learn and discover them. Topology: Walk between Intuition and abstraction recently, the time to read again the point set topology (points set topology), this is my third re-study of the theory. I watch TV dramas and novels, rarely have the interest to see the second time, but, for mathematics, every look has a new inspiration and harvest. Algebra, analysis, and topology, are known as the three pillars of modern mathematics. The initial read topology is in twoThree years ago, due to the need of learning manifold theory. However, with the accumulation of knowledge, it is found to be the foundation of many theories. It can be said that there is no topology, there is no modern meaning of analysis and geometry. The most basic concepts we have access to in various branches of mathematics, such as limits, continuity, distance (measure), boundary, and path, are derived from topologies in modern mathematics. Topology is a fascinating discipline that links the most intuitive phenomena with the most abstract concepts. Topology describes the concepts of universal use (such as open set, closed set, continuous), which we take for granted, but, indeed, to define it requires the deepest insight into their nature. After a long period of hard work, mathematicians got a modern definition of these concepts. Many of these first-glance looks are surprising-how can they be defined like this. The first is open set. When we study elementary mathematics, we all learn to open the interval   (A, B). However, this is only in one line, how to promote to two-dimensional space, or higher dimensional space, or other forms of it? The most intuitive idea is "a set that does not contain boundaries." However, the question comes, give a set, what is the "boundary"? In topology, open set is the most fundamental concept, which is defined on the basis of set operations. It requires the open set to conform to the condition that the arbitrary and finite intersections of open sets are still open sets. When I was first, I really baisibuxie the way it was defined. However, after reading, seeing and doing a lot of proofs, it was found that the definition of such a very important meaning is: it ensures that each point in the open concentration has a neighborhood contained within this set-all points are kept away from the outside (complement set). Such an understanding should have a clearer geometrical meaning than the definition of using set operations. However, intuitive things are not easy to directly form a rigorous definition, the use of set operations is more stringent. In the definition of set operation, the closeness of arbitrary aggregation is the intrinsic guarantee of this geometrical characteristic. Another example is "continuous functions" (continuous function). When learning calculus, a familiar definition is "for any epsilon > 0, there is Delta > 0, which makes ...." ", the most intuitive meaning behind it is" near enough points to ensure that they are mapped to any small range. " However, epsilon, Delta is dependent on the real space, not the real space mapping and how to do? The definition of a topology is "if the original image of any open set in a mapped domain is open, then it is contiguous." "There's nothing epsilon here." "The original image of open set is open set" The key here is that in topology, the most important meaning of open set is to convey the meaning of "neighborhood"--the opening set itself is the neighborhood of the included points. This is a logical way to define the continuum. To adjust the argument a little bit, the definition above becomes "for any neighborhood U of f (x), there is a neighbor of X."Domain V, so that the points inside V are mapped to U. "In this case, we can feel why the open set has a fundamental meaning in topology." Since the open set conveys the meaning of "neighborhood", its most important function is to express which points are relatively close. Give a topological structure, which is to point out which is open set, and thus point out which points are closer, thus forming a aggregation structure-this is the topology. But this can also be described by distance, why use open set it, but not intuitive. In a sense, the topology is "qualitative", distance measurement is "quantitative". As the continuous deformation, the distance will change, but the close point is still close, so the inherent topological characteristics will not change. The topological study is the intrinsic characteristic-invariance in continuous change. In the basic concept of topology, the most puzzling is the "tightness" (compactness). It describes a space or a set of "tight". The formal definition is "if any of the open overlays of a set have finite sub-overrides, then it is tight". At first glance, it's a little confusing. What exactly does it want to describe? And the "tight" the adjective and how to relate it? An intuitive point of understanding, a few sets are "tight", that is, unlimited points scattered in, it is not possible to fully diffuse. No matter how small the neighborhood, there must be an infinite number of points in the neighborhood. The mystery of this definition of compactness is in limited and infinite transformations. A tight collection is covered by an infinite number of small neighborhoods, but the finite one can always be found to cover the whole. So what are the consequences? An infinite number of points scattered in, there is always a neighborhood wrapped countless points. The neighborhood is so small-this guarantees that there are limits in the infinite sequence. Although the concept of compact is a bit less intuitive, it plays an extremely important role in the analysis. Because it relates to the existence of limits-this is the basis of mathematical analysis. The friends who understand functional analysis know that the sequence is convergent, and many times it is seen. In calculus, an important theorem-bounded sequences necessarily contain convergent sub-columns, which is the root of this. Before learning the topology, or other modern mathematical theories, our mathematics has been in the limited dimensional Euclidean space, that is a perfect world, with all the good properties, Hausdorff, locally compact, Simply connected,completed, There is also a set of linear algebraic structures, with well-defined measures, norms, and inner product. However, with the deepening of research, after all, still have to go out of this circle. At this time, things that were taken for granted have become less inevitable.                      two points must be able to separate? You have to prove that space is Hausdorff.                 There must be a limit to the number of      bounded series. This is only in the locally compact space.                      any two points in a continuous body must have a path connection? This is not necessarily. Everything seems counterintuitive, and it does exist. From linear algebra to general groups, from finite dimensions to infinite dimensions, from metric space to topological space, the whole understanding needs to be re-cleaned. Moreover, these are not merely mathematicians ' concept games, because our world is not a finite-dimensional vector that can be fully expressed. When we look at things that are not vectors that can be expressed, the concepts of measurement, algebra, and analysis must be re-established, and the starting point is in the topology.

Several math books recommended by Lindahua