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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.

57719 questions
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Why are clopen sets a union of connected components?

The wikipedia page on clopen sets says "Any clopen set is a union of (possibly infinitely many) connected components." I thought any topological space is the union of its connected components? Why is this singled out here for clopen sets? Does it…
Cellar Door
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Closed sets as compact sets

While playing around with some basic general topology, I have thought of some problems whose solutions are not so obvious (at least to me), and surprisingly I do not remember having seen these anywhere. Disclaimer #1 : I have already posted those…
D. Thomine
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If the composition of two maps is continuous and one of the maps is also continuous, then is the other continuous

Let $f:X\to Y$ and $g:Y\to Z$ be maps of topological spaces. Assume that the composition $g\circ f$ is continuous and that $f$ is continuous. Is $g$ necessarily continuous? If this is not true in general, is it true under some hypotheses on $X$,…
Hannah
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Is $(\mathbb{R}^2, lexicographic) \cong (\mathbb{R}, discrete)\times (\mathbb{R}, usual)$?

I have the following question: Is $(\mathbb{R}^2, lexicographic) \cong (\mathbb{R}, discrete)\times (\mathbb{R}, usual)$? Note: the symbol $\cong$ denotes a homeomorphism between topological spaces What I think: I think that in fact there is a…
LFRC
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Is $[0,1]^\omega$ a continuous image of $[0,1]$?

Is $[0,1]^\omega$, i.e. $\prod_{n=0}^\infty [0,1]$ with the product topology, a continuous image of $[0,1]$? What if $[0,1]$ is replaced by $\mathbb{R}$? Edit: It appears that the answer is yes, and follows from the Hahn-Mazurkiewicz Theorem (…
Iian Smythe
  • 1,289
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If a subset of the plane has open intersection with every line, is it open?

Suppose that $U \subset \mathbb{R}^2$ is such that $U \cap L$ is open in $L$ for any line $L \subset \mathbb{R}^2$ where $L$ inherits the subspace topology from $\mathbb{R}^2$ (ie. $L \cong \mathbb{R}$). Does it follow that $U$ is open? I keep…
Mike F
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Other awesome topology related videos like this one?

Turning a sphere inside-out: http://www.youtube.com/watch?v=BVVfs4zKrgk And part 2: http://www.youtube.com/watch?v=x7d13SgqUXg This is really, really cool. They describe things in really simple terms though (like referring to curvatures with smiles…
Steven Lu
  • 391
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Lower Limit Topology?

Show $(0,1)$ is open but not closed in the Lower Limit Topology. I know that $[a,b)$ is open and closed in the lower limit topology, but I am not sure how to prove this one. Thanks for any help.
kumhmb
  • 135
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Uncountable family of uncountable compact subsets of $\mathbb{R}$

Does there exists an uncountable family $\mathcal{A}$ consisting of uncountable compact subsets of $\mathbb{R}$ and pairwise disjoint?
user10
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Help in proof of Compactness implies limit point compactness.

Munkres' Topology says Theorem 28.1. Compactness implies limit point compactness, but not conversely. Proof. Let $X$ be a compact space. Given a subset $A$ of $X$, we wish to prove that if $A$ is infinite, then $A$ has a limit point. We prove…
Daniel Donnelly
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Showing the Sorgenfrey Line is Paracompact

The Sorgenfrey Line is $\mathbb R_/ = (\mathbb R, \tau_s)$ where $\tau_s$ is the topology on $\mathbb R$ with base $\{[a, b)\ |\ a, b \in \mathbb R\}$. I know how to show $\mathbb R_/$ is not locally compact. It turns out that the only compact sets…
Geoff Pointer
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For which topological spaces does the closure commute with the interior?

For which topological spaces $X$ is it the case that for all $Y \subseteq X$, $\mathrm{cl}(\mathrm{int}(Y)) = \mathrm{int}(\mathrm{cl}(Y))$? If one replaces equality with inclusion (i.e., $\mathrm{cl}(\mathrm{int}(Y)) \subseteq…
Geoffrey Trang
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Union of ascending chain of Topologies

Let $\lbrace \tau_n:n\in \mathbb{N}\rbrace$ be an ascending chain of topologies on a nonempty set $X$. Then is $\bigcup\limits^{\infty}_{n=0}\tau_n$ a topology on $X$? I have a strong notion that it may not be a topology. But I am not getting the…
Anupam
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Finding a counterexample; quotient maps and subspaces

Let $X$ and $Y$ be two topological spaces and $p: X\to Y$ be a quotient map. If $A$ is a subspace of $X$, then the map $q:A\to p(A)$ obtained by restricting $p$ need not be a quotient map. Could you give me an example when $q$ is not a quotient…
Xena
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How many sets can we get by taking interiors and closures?

Consider the following problem. I am looking for the maximum number of different sets that we can generate by one set $B \subseteq \mathbb{R}$ by taking a finite amount of closures and interiors. For example $\{0\}$ generates the sets $\{0\}$ and…
Michiel Van Couwenberghe
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