0
votes
0answers
10 views

Criteria to judge te quality of a journal [migrated]

Is a journal with high impact factor is better than that with low impact factor? Is impact factor is right criteria to judge the quality of a journal?
5
votes
4answers
509 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
87 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 ...
1
vote
0answers
37 views

Soft question: is there a generalization of compactness satisfying these conditions?

A good intuition about compact topological spaces is that they're spaces that aren't missing any points (not really a huge fan of the "compact = small" intuition, but that's another story). ...
2
votes
2answers
75 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 ...
6
votes
3answers
171 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
131 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
41 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, ...
0
votes
1answer
81 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 ...
11
votes
2answers
162 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
142 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
228 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
522 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
141 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
169 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 ...
27
votes
7answers
816 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
87 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
132 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
218 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
105 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
70 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 ...
63
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
736 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
98 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
239 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
212 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 ...
28
votes
7answers
882 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
111 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 ...
8
votes
2answers
440 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
42 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
212 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 ...
7
votes
2answers
157 views

What is the relationship between completeness and local compactness?

On the one hand, $\mathbb{Q}$ is neither complete (as a metric space) nor locally compact (as a topological space). On the other hand, $\mathbb{R}$ is both complete and locally compact. My question ...
5
votes
2answers
177 views

The complement of a torus is a torus.

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can ...
4
votes
3answers
200 views

Interesting theorems/facts about identification spaces

I am now studying algebraic topology (still at the beginning). I am now studying identification spaces, adjunction spaces,... As I still don't know how these concepts are going to be used, I think I ...
2
votes
1answer
28 views

Is every GO-space collectionwise normal?

Is every GO-space collectionwise normal? And what is the relation between collectionwise normal and monotonically normal? I know a GO-space is always monotonically normal. Thanks.
0
votes
1answer
40 views

Understanding the relation between countably paracompact and monotonically normal

Does monotonically normal imply countably paracompact? Thanks ahead:)
6
votes
1answer
128 views

Is this an interesting generalization of the notion of an open set?

Let $X$ denote a topological space. Some subsets $A \subseteq X$ might have the property that $\partial A = \partial(\mathrm{int}\,A).$ This is certainly true if $A$ is open (since open implies ...
3
votes
0answers
117 views

Why don't we introduce the concept of base for a topology in a minimal way?

Why don't we introduce the concept of base for a topology in a minimal way exactly as we did in Linear Algebra? Edit: A topology can be obtained from a base by considering all possible unions of the ...
5
votes
3answers
1k views

Topology Prerequisites for Algebraic Topology

Note: There is another question of the same title, but it is different and asks for group theory prerequisites in algebraic topology, while i want the topology prerequisites. I am a physics ...
10
votes
2answers
246 views

Does there exist another way of obtaining a topological space from a metric space equally deserving of the term “canonical”?

Every metric space is associated with a topological space in a canonical way. According to this source, this amounts to a full functor from the category of metric spaces with continuous maps to the ...
9
votes
4answers
544 views

How can I explain topology to my grandmother?

I was recently look at a post on tex.stackexchange about explaining $\LaTeX$ to the OP's grandmother. I was wondering, could the same thing be done for topology? Except in this case the "grandmother" ...
7
votes
1answer
116 views

Ultrafilters in Topology

I have seen the use of ultrafilters in topology to prove Tychonoff's theorem. In fact, this is the only application of ultrafilters in topology I have been able to find so far. Are there any other ...
1
vote
1answer
113 views

Intuition Regarding Homotopic Spaces

I am just starting to do some algebraic topology (very basic stuff) so have obviously just been introduced to the notion of homotopys, contractible spaces and homotopic spaces. It's this last one that ...
73
votes
3answers
8k views

Topology: The Board Game

Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website with a Creative Commons Attribution 4.0 license. ...
6
votes
3answers
219 views

Model Theory and Topology Connections

I have studied a bit of model theory, when I say "a bit" I have studied much more than is available to a typical undergraduate in the UK (i think, certainly from what I have seen) but I am sure this ...
4
votes
1answer
171 views

On convergence of nets in a topological space

Let's consider a topological space that is not necessarily metrizable. Question: I wonder what is the motivation for defining convergence of nets in a topological space? What do we gain in working ...
6
votes
2answers
262 views

Has the Four-Color Theorem been accepted?

The Wikipedia article is ambivalent about this, stating "...Since then the proof has gained wider acceptance, although doubts remain". The MathWorld entry isn't much more reassuring. This is ...
15
votes
1answer
447 views

Given a group $ G $, how many topological/Lie group structures does $ G $ have?

Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have? Any abstract group $ G $ will have the structure of a discrete topological group ...
9
votes
2answers
263 views

What are some motivating examples of exotic metrizable spaces

Among topological spaces, the metric spaces are usually considered to be the tame animals. Describing the topological notion of closeness by a distance is so intuitive (as opposed to the abstract ...
8
votes
2answers
323 views

Fast paced book in point-set topology to move on to algebraic topology

I am sorry, if this is a repetition of previous questions. But my case is sightly different. I am a physics undergrad who wants to shift to pure maths, and I want to study topology. The supreme ...