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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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Is empty set always part of a basis of a topology?

I am reading about point-set topology. Is empty set always part of a basis of a topology?
hereitis
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Three-point compactifications of $\mathbb{R}$?

A few aeons ago I read a proof that the real line has no $n$-point compactifications for $n\ge 3$. I don't remember it. Could someone remind us all? This says in effect that a function $f:\mathbb R\to\mathbb R$ cannot have more than two horizontal…
Michael Hardy
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Clopen sets of an infinite set with cofinite topology

Let $X$ be an infinite set with its cofinite topology. I was asked to mention the clopen sets. I think the only clopen sets are X and the empty set, because otherwise, if A is a clopen set, it would mean that both A and its complentary set are…
Ramona Ioana Mihu
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For a finite complement topology , to which point or points does the sequence converge?

For a finite complement topology on $R$ to which point or points does the sequence ${\frac{1 }{n}}$ converge? For a finite complement topology on real numbers the only set including the limit point of sequence is $R$ so the sequence converges to…
Kavita
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Is there a continuous function from the torus to the sphere? or from sphere to the torus?

I know that a sphere and a torus are not homeomorphic. But exists a continuous function from the torus to the sphere? or to be not homeomorphic implies that they are imposible (continuous function)
Joe
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topology - Quotient topology

Let $k:X \to Y$ be an onto map.How to prove that the quotient topology on $Y$ induced by $k$ is the largest topology relative to which $k$ is continuous.
ccc
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Show that $G/H$ has the trivial topology if $H \subset G$ is dense

I'm having trouble on this assignment question and was hoping someone can point me in the right direction. Let $G$ be a group equipped with a Hausdorff topology in which the group operation and inversion are continuous. Let $H$ be a subgroup. Let…
Kevin Hsu
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Dimension of open subsets of $R^n$

Does an open subset of $R^n$ exist that has dimension less than $n$ in the standard topology? All the less than $n$ dimensional subsets I can think of are not open.
MeMyselfI
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If $ A$ is open and $ B$ is closed, is $B\setminus A$ open or closed or neither?

If $ A$ is open and $ B$ is closed, is $B\setminus A$ open or closed or neither? I think it is closed, is that right? How can I prove it?
Jill Clover
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Compact space which is not sequentially compact

Greets So this is exercise 17G.1 of Stephen Willard's "general topology", and it's stated: Show that there is a compact space that is not sequentially compact [Hint: Consider an uncountable product of copies of $[0,1]$]
Camilo Arosemena-Serrato
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Can the closure of a simply connected domain in the complex plane fail to be simply connected?

Is it possible for the closure of a simply connected domain in the complex plane to not be simply connected? Intuitively it seems the closure is simply connected but I can't prove it. Is it enough to show that every point is homotopic to some point…
Mykie
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It is possible to prove that these two collections generate the same topology on $ \mathbb{X} $?

Let $ \mathbb{Y} $ a topological space whose topology $ \tau $ is generated by a collection of subsets $ \mathcal{B} \subset 2^\mathbb{Y} $. In other words the topology $ \tau $ of $ \mathbb {Y} $ is the smallest topology ("Smaller" in partial order…
Elias Costa
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Why axiom 3) of topology is redundant?

I'm learning topology and I'm reading Dugundji's General Topology book. He gives the definition of axioms of a topology $\tau$ on $X$: 1) & 2) (a topology is closed under arbitrary union and finite intersection ) 3) $\emptyset$ and $X$ belong to…
HeMan
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A connected space which is neither locally connected nor path connected

This is Problem 3.19.3 of Dieudonné's Foundations of Modern Analysis (in my words). For $x$ a rational number, let $E_x=\{x\}\times\left[-1,0\right[$, and for $x$ an irrational number, let $E_x=\{x\}\times[0,1]$. Let $E=\bigcup_{x\in\mathbf{R}}E_x$…
Stefan
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The topological boundary of the closure of a subset is contained in the boundary of the set

Let $X$ be a topological space and $N$ a subset of $X$. I want to show that $\partial \bar N\subset \partial N$. I know that since $\bar N$ is closed then $\partial \bar N\subset \bar N$. By definition $\bar N=N \cup \partial N$, now if $x\in…
palio
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