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Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

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Saveliev Illustrated Topology question on locked hands

Exercise 1.4 on Saveliev "Illustrated Topology" gives a picture of locked hands and asks how can we unlock them without breaking the loops. I've thought about it and couldn't find a way. I'd appreciate a hint. Is it even possible?
blz
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Structures on spaces of topologies

Does there exist a fruitful notion of "moduli space of topologies"? For example, is it possible to define useful/natural topologies on the set of topologies on a given set $A$? When does it make sense to talk about the convergence of a sequence of…
merle
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Open cover of $\mathbb C$

Today one of my students asked an interesting question, which I was unable to answer. Concretely: Let $(z_n)_{n\in \mathbb N^+}$ be an enumeration of $\mathbb Q+\mathrm i\mathbb Q$. The question is, whether $(B_{1/n}(z_n))_{n\in \mathbb N^+}$ is a…
Nico Zimmer
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Covering for connected and locally path connected spaces

Under the condition that the spaces (or maybe just the total) are connected and locally path connected, is then the a covering the same as a homeomorphism?
Down
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$A$ closed in $Y$ and $Y$ closed in $X$ , then $A$ is closed in $X$

This sounds really simple and I'm struggling with it. I first tried to show that $X-A$ had to be closed by trying to show the complementary had to be open (trying to express it as union or intersection of known opens), but I couldn't do it: $(X-A)$…
MyUserIsThis
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If product of two sets $A\times B$ is closed, are $A$ and $B$ closed?

If $A\times B$ is closed in $X\times Y$, then are $A$ and $B$ closed in $X$ and $Y$ respectively?
Anupam
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non sequential spaces

I am looking for a space which is not sequential. I tried to build this example: Take $X$ to be a countable union of the folloing points: $X= (\bigcup_{n=1}^\infty( \bigcup_{k=1}^\infty…
topsi
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Does subspace topology affect the compactness?

Consider the function $f(x)=x+2$. Let the domain be $\left(0,1\right)$, then the range is $\left(2,3\right)$. $\left(0,1\right)$ is not compact in $\mathbb{R}$ since it is not closed in $\mathbb{R}$. But if I consider the subspace topology with…
Ypbor
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about the definition of connected sets in $\mathbb R^n$

(Edited to make reference to the topological space $X$ precise.) A subset $A$ of a topological space $X$ is connected if there are no two open subsets $O_1$ and $O_2$ of $X$ such that (a) $A \subseteq O_1 \cup O_2$, (b) $A \cap O_1 \neq \emptyset$,…
Eilon
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Do there exist continuous bijections from Euclidean space which change the topology?

Do there exist continuous bijections from Euclidean space $X: \mathbf R^n\to M$ whose inverse is not continuous (where $M$ is a $n$ dimensional manifold)? I'm aware of continuous bijections from subsets of Euclidean space which change the topology,…
dennis
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topological KC - space

A topological space X is KC – space if every compact subsets are closed. question: Does a KC - space contains a minimal KC topology?
Alireza
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Why is the subspace topology defined as it is?

Given a topological space $(X, \tau)$, and $Y \subset X$, the subspace topology on $Y$ is defined to be $$\tau_Y = \{Y \cap U : U \in \tau\}.$$ The definition itself is clear and self-explanatory, what I am wondering here is why is it defined this…
CBBAM
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Finding limit points of subsets of the cofinite topology on $\mathbb{Z}$

Is my reasoning correct? Problem: Let $(Z,\tau)$ be the cofinite topology on $Z$. Find the limit points of the sets: $A = \{1,2,\dots,10\}$ $E$, the even integers My solution: $A$ is closed (finite), so it contains its limit points. Let $x$ in…
mrk
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Pre-images of closed sets are open

Let $X$ and $Y$ be two topological spaces and let $f$ be such a map that $f^{-1}(A)$ is open in $X$ for any closed $A$. Note that if $X\stackrel{f}{\longrightarrow}Y\stackrel{g}\longrightarrow Z$ are two such maps, then $g\circ f$ is continuous.…
SBF
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Topology exercises - closure, frontier

Are my proofs correct? The topology is $\mathbb{R^n}$ Exercises. Prove that a set $A$ is closed iff $Fr(A)\subseteq A$ A set $A$ is closed iff $A=Cl(A)$ For any set, $A$, $Fr(A)$ is closed For any set $A\subseteq \mathbb{R^n}$,…
mrk
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