# Tagged Questions

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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### Topological Groups and the Mapping Class Group

I am currently studying mapping class groups. In particular, I am looking at a relation between the group of topological automorphisms of a topological group (i.e. group automorphisms which are also ...
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### Correct definition of a regular covering without global connectedness hypotheses

Let $p:Y\to X$ be a covering map of topological spaces where $X$ is assumed to be locally path connected (and hence the same is true of $Y$) but neither $X$ nor $Y$ is assumed to be connected. In this ...
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### In a complete metric space $(X,\rho)$, show that if $E$ and $X\setminus E$ are dense, then at most one of them is a countable union of closed sets.

The problem statement is in the title. I approached this proof using contradiction. My attempt was: Suppose that both $E$ and $X\setminus E$ are dense and that both are a countable union of closed ...
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### Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
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### Difference between Metric Space and Topological Space

I am reading Chapter 11 of Real Analysis written by Royden and Patrick (4th). It says "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise ...
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### Find Group Action

I'm asked to find a group action G on the unit cylinder C such that $C/G$ is homeomorphic to the torus. Would $\pm1 \cdot (x,y,z) = (x,y,\pm z)$ work? The only problem here is that G should only act ...
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### Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
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### When is a locally metrizable space actually metrizable?

A topological space is metrizable when there is a metric that induces the topology. A topological space is locally metrizable when every point has some neighbourhood that is metrizable. According to ...
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### What are the names for the structures obtained when we drop some topological space axioms?

Motivation: If I start with the group axioms and drop the requirement that I have inverses, I get the monoid axioms. If I proceed to drop the requirement that I have an identity, I get the semigroup ...
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### Proof of “invariance of domain theorem”

How can I get a proof of "invariance of domain theorem"? Of course, I would like to know brief explanation. Now, this statement is here ; http://en.m.wikipedia.org/wiki/Invariance_of_domain . Thank ...
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### Morse-Smale Complex, boundary on the number of segments by the number of critical points.

I am looking for a known upper bound on the number of monotone regions of a Morse function by the number of its critical points in the interior of the manifold and on its boundary. Here I try to ...
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### Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
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### A domain on a sphere is simply connected if and only if its complement is connected

I think the statement that a domain (open connected set) in a sphere is simply connected if and only if its complement is connected is a standard result. But how can one prove it? Is it possible to ...
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### Locally / Strongly connected product spaces

I have to give necessary and sufficient conditions s.t. $X=\prod_i X_i$ is locally / strongly connected, where $(X_i,\tau_i)$ are non-empty top. spaces. First locally: Assume all $X_i$ are locally ...
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### Set of limit points of S is closed in a metric space X

A point $x \in X$ is a limit point of a subset S of X, if every ball $B(x;\varepsilon)$ contains infinitely many points of S. Show that x is a limit point of S iff there is a sequence {$x_{j}$} in ...
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### Exploiting the compactness of the unit circle to prove the following proposition.

I am trying to prove that a locally convex topological vector space is equivalent to a semi-normed topological vector space. I have worked through the proof but I am unsure because of the following ...
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### How to prove in a topological vector space in R^n: cl(A) + cl(B) =cl(clA+clB), where cl denotes closure and A, B compact convex sets?

Let A,B be compact convex sets, then Is equal to cl(A+B)=cl(cl(A)+cl(B))? why? Thank you.
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### An exercise from the handbook of set-theoretic topology

This is an exercise from the handbook of set-theoretic topology (Exercise 13.3): Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact ...
I have this question below and am not sure if my approach is correct. Can anyone please advise me? Thanks. Question: Let $f:\mathbb{R}^n \to \mathbb{R}^m$ and suppose there is a positive constant $K$ ...
Let us call a set in a topological space finitely decomposable set (FDS) iff it can be rewritten using the standard set operations $\cup$ and $\sim$ and only finitely many open sets. I'm looking for ...