# Tagged Questions

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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### Understanding the relation between countably paracompact and monotonically normal

Does monotonically normal imply countably paracompact? Thanks ahead:)
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### 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 ...
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### 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 ...