4
votes
1answer
38 views

Is the “product rule” for the boundary of a Cartesian product of closed sets an accident?

Given two closed sets $A$ and $B$ living in topological spaces $X$ and $Y$, the boundary of $A\times B$ in the product topology, denoted (suggestively) by $\partial(A\times B)$, is given by ...
0
votes
4answers
36 views

Explaining what is Pathwise-connectedness.

I'm an average guy but interested in explaining myself maths through illustrations and intuition(which at times fails!!!).I'm preparing myself for my calculus of several variables exam. I'm studying ...
5
votes
0answers
47 views

How to master general topology for analysis?

I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. The first big ...
1
vote
0answers
24 views

Continuous maps vs. open maps [duplicate]

When we study topology, we typically study topological spaces and continuous maps between them. From a categorical perspective, this is "wrong," because continuous maps are not the structure ...
2
votes
1answer
58 views

How to conceptualize unintuitive topology?

I found Project Origami: Activities for Exploring Mathematics in my university's library the other day and quickly FUBAR'd (folded-up beyond all recognition) the couple sheets of paper I had with me ...
7
votes
3answers
127 views

Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V ...
1
vote
0answers
44 views

Why are empty measurable spaces and empty topological spaces not desirable?

The definition of a $\sigma$-field $\mathscr{F}$ on a set $X$ (or $\sigma$-ring) requires $\mathscr{F}$ to be a non-empty subset of $\mathscr{P}(X)$. Why is this convention taken? What is the issue ...
6
votes
2answers
163 views

Why should the generalization of a 'sequence' be called a 'net'?

The title says it all, really. Reading through Reed & Simon's book on functional analysis, I have now reached the chapter on topological spaces, and the notion of a net is introduced there to ...
4
votes
1answer
135 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
4
votes
1answer
80 views

Whatever Happened to Nearness Spaces?

I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange ...
4
votes
0answers
110 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
3
votes
0answers
37 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
3
votes
1answer
92 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
1
vote
1answer
39 views

A special kind of metric-spaces

Is there a special name for those metric-spaces or topological spaces in which every non-empty open set is uncountable ?
1
vote
1answer
68 views

Sphere homeomorphic to plane?

I just took a course in general topology about a month back, and I was wondering whether it was possible to explain why the Earth seems flat from our point of view but is in fact a sphere using the ...
2
votes
1answer
37 views

A soft question on the dimension of normed spaces

There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional. An example is of course the compactness of closed bounded sets. Another ...
3
votes
3answers
184 views

How to explain why free loop space, or based loop space, is infinite dimensional to non-math people.

I am giving a math talk to non-mathematicians. I was wondering how to explain how the free loop space, or based loop space, of a topological space is infinite dimensional so that a non-mathematician ...
17
votes
1answer
479 views

Why do we study Polish spaces?

In descriptive set theory, a lot of space is devoted to properties of Polish spaces. (A Polish space is a topological space, which is separable and completely metrizable.) I would like to know why ...
2
votes
3answers
67 views

How 'normal' are normal spaces?

Is normality a property that is 'easily' satisfy by a given space? I mean by this: are non-normal spaces hard to construct/unnatural compared to normal spaces?
1
vote
1answer
87 views

What path to get to topology?

I am in calc 1 right now and was wondering what kind of journey is ahead of me before topology. I really want to study high level math like this but am not sure if I want to major in math. I am a pre ...
6
votes
1answer
107 views

difference between sequence in topological space and metric space

Reading the book about topology, I find an interesting difference between two spaces: We use net convergence $\{p_\lambda\}_{\lambda\in\Lambda}\rightarrow p$ in frontier, but use sequence ...
6
votes
4answers
728 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
5
votes
1answer
125 views

Struggling with Topology. Any advice?

I'm a Junior Mathematics major at a small Liberal Arts college. I'm currently taking first semester topology (Munkres text). I feel like I'm barely able to tread water in this course. I was able to ...
2
votes
2answers
78 views

Algebraic Object in Topology

Common algebraic categories are group, ring, module and algebra. Some of them have the corresponding object in topology, like topological group and topological linear space. We define them by making ...
7
votes
3answers
285 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
3
votes
1answer
154 views

Are continuous mathematical models of discrete physical phenomena messy because of a disconnect between “continuous” and “discontinuous”?

Examples from statistical mechanics and continuum mechanics abound: a discrete phenomenon (e.g. kinetic energy of molecules) is "averaged" out over the constituents of the system to which it applied ...
0
votes
2answers
52 views

Metric Topology

Suppose our topological space is $\mathbb{R}$. Why is each basis element $(a,b)$ for the order topology is a basis element for the metric topology? Munkres says, ...
1
vote
1answer
99 views

What is the contribution of group theory to topology?

An answer for a question on MathOverflow.net which asked for some recommendations on textbooks for books in topology received the following comment: "It's a great book to introduce applied ...
12
votes
2answers
186 views

How can you describe topology to a non-mathematician without using continuous deformations?

One of the most frequently used ways to describe topology to non-mathematicians is that it studies the properties of objects that are preserved under deformations where ripping or tearing is not ...
7
votes
1answer
160 views

General Topology: “Follow your Nose Approach”

So, this is definitely a soft question and I apologize. I've been in point set topology for about a week and I have two questions, Everything spews from definition, so should I dismiss my geometric ...
13
votes
2answers
282 views

Motivation for introducing algebraic topology?

What kind of topological questions does algebraic topology answer where point set topology is not enough? Phrased differently: Where is the line (or maybe intersection) between point set topology ...
7
votes
3answers
580 views

Why are compact sets called “compact” in topology?

Given a topological space $X$ and a subset of it $S$, $S$ is compact iff for every open cover of $S$, there is a finite subcover of $S$. Just curiosity: I've done some search in Internet why compact ...
6
votes
3answers
171 views

How much topology for graph theory?

I am writing a thesis in the context of descriptive complexity in theoretical computer science and therefore need to study a little bit of graph theory. My background is not mathematics but computer ...
13
votes
1answer
183 views

What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
26
votes
6answers
940 views

How to develop intuition in topology?

Is there any efficient trick (besides doing exercises) to develop intuition in topology? The question is general but i would like to add my view of things. I started to teach myself topology through ...
0
votes
1answer
91 views

How to make a ghost manifold [closed]

How does one mathematically define a manifold that can pass through another manifold? A "ghost" passing through a "wall" type construction. I understand that this may be done by creating a copy of the ...
0
votes
3answers
175 views

A little confusion about compactness and connectedness

This question may be a bit simple or even naive for some people but it indeed confuses me for a long time. Thank you all if you provide any explanation. I know concepts: compactness means any open ...
7
votes
1answer
271 views

What is the significance of limit points?

When I had my first taste of topology a couple of years ago, our lecturer emphasized the following notions. closed set, closure, closure point open set, interior, interior point Of course, these ...
2
votes
0answers
109 views

Is my general approach to proofs acceptable? A general topology example.

Proving: $A$ is closed iff $A = \bar{A}$. "To the right": If $A$ is closed, $ A = \bar A$ If $A$ is closed this means that it contains all of its own accumulation points. And we would find that its ...
1
vote
0answers
77 views

Importance of Banach fixed point in mathematics

I know the proof of Banach fixed point theorem in complete metric space and two example of it. It is useful to prove Picard's theorem on existence and uniqueness of ordinary differential equation. In ...
65
votes
12answers
5k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
14
votes
4answers
795 views

Can we get un-obvious results by defining sophisticated topologies?

What I originally found so interesting about general topology was how general a type of thing a topology is, and how the terminology open, closed, compact, continuous, convergence et cetera means ...
3
votes
2answers
105 views

difference between $\mathbb{R}^2$ and $\mathbb{R} \times \mathbb{R}$

I was going through some of notes in regards to Fourier analysis and I noticed that in some cases when dealing with a 2 dimensional transform the function $f \in \mathbb{R}^2$ while other times $f \in ...
8
votes
1answer
270 views

Is the number 8 special in turning a sphere inside out?

So after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here? ...
3
votes
2answers
274 views

What is the difference between differential topology and calculus on manifolds?

I'm trying to teach myself one and bought a book on the other. It seems to me that they both cover about the same material. This leads to the question: What is the difference between differential ...
31
votes
7answers
1k views

Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed ...
4
votes
3answers
114 views

What mathematical objects permit “taking of limits”?

Background I have been reading a lot of abstract algebra recently (at the level of Artin/Dummit & Foote/Herstein Topics in Algebra for those of you familiar with these books). I have noticed ...
9
votes
2answers
609 views

How to deal with Homeomorphisms?

I have one doubt that may be too general, I don't know, so sorry if this is not a good place to ask it. I've also seem many other people with the same problem that I have, so I think that if this ...
1
vote
0answers
43 views

How to build the largest sambusa

I was making sambusa last night. Typically when mama cooks them she has small circles of dough, but mama is not here so I went to the store and bought the dough, and it came in the shape of a ...
8
votes
1answer
239 views

what's the role of fiber bundles play in understanding the base space?

Suppose $\pi: E\to B$ be a fiber bundle, and assume $B$ is connected, I want to know: 1, What do the fiber bundles (e.g., the covering spaces) on $B$ help understand the topology, or the structure on ...