# Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

2,951 questions
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### Is the realization of a proper map of simplicial spaces proper ?

Let $f:X \rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from ...
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### Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
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### Colimits in the category of smooth manifolds

In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of ...
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### Is a proper quotient map closed ?

I am trying to produce closed quotient maps, as they allow a good way of creating saturated open sets (as in this question). A map $f:X\rightarrow Y$is called proper, iff preimages of compact sets ...
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### how slow can the dimension of a product set grow?

Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$: $\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$, where $\sim$ denotes "...
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### Continuous function from $[0,1]$ to $[0,1]$

Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
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### Is a inverse limit of compact spaces again compact ?

Then one can construct a model for the inverse limit by taking all the compatible sequences. This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces ...
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### What is enough to conclude that something is a CW complex?

This question was something I considered when looking into CW-structures on Grassmannians, but I found no general treatment of this in the literature: Question: Assume that $X$ is an $n-1$ ...
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### SU(2) representations of alternating knot groups

Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...
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### Is there a non-trivial topological group structure of $\mathbb{Z}$?

More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?
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### What is a reference for profinite sets?

The question is in the title. The motivation behind the question is as follows: there are plenty of references about profinite groups and profinite completions of groups. It seems that their not ...
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### Using topology to characterize embedded Lie subgroups of Lie groups.

Cartan's theorem states that any topologically closed subgroup of a Lie group is an embedded Lie subgroup. This leads us to ask the following question: Can we replace "topologically closed" with a ...
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### Can a connected planar compactum minus a point be totally disconnected?

What the title said. In a slightly more leisurely fashion:- Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be ...
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### Why is it useful to classify the vector bundles of a space?

It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
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### Intersection form in twisted homology (homology with local coefficients)

The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
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### Which Fréchet manifolds have a smooth partition of unity?

A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is: Which Fréchet manifolds have a smooth partition of unity? How is the ...
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### How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
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### Confusion over a point in basic category theory

"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
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### Cryptomorphisms

I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are Topological Spaces. These can be defined in terms of open sets, ...
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### “Category” of Nonempty Metric Spaces and Contractive Maps?

The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A \to B$ such that $d_B(f(a), f(a')) \leq d_A(a, a')$) as ...
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### Simple question of topological cofibration

I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
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### Which spaces are inverse limits of discrete spaces ?

There is the following theorem: "A space $X$ is the inverse limit of a system of discrete finite spaces, if and only if $X$ is totally disconnected, compact and Hausdorff." A finite discrete space ...
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### Cone in a metric space

Hi everybody, We know the definition of a cone in a Real Banach Space. I want to know if there is any definition for a cone in an abstract metric space. Have you ever seen such definition anywhere? ...
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### definition of the end of a manifold?

Hey everybody, I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function asigning to each compact set K a conected component e(K) of the ...
Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$. Let's call the following statement (2): ...
I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes: Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...