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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 the interior of the closure of the interior of the closure of a set equal to the interior of the closure of that set?

Let $S$ be a subset of a topological space. I want to prove or disprove the following claim: $\left(\overline{\left( \overline{S} \right)^\circ}\right)^\circ=\left( \overline{S} \right)^\circ$ Setting $A=\left( \overline{S} \right)^\circ$, we have:…
A-B-izi
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Can you add a single point to any completely regular (not normal) space to get a normal space?

This is a follow up to my previous question (unfortunately closed as a duplicate). There the problem was to turn the Moore plane into a normal space by adding a single point. Brian M. Scott gave an answer to this specific problem years ago. This…
Alice Munro
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Compactness properties of rational sequence topology

For each irrational ${x}$ we take a sequence of rational numbers ${x_k}$ with the property that ${x_k}$ converges to it in the Euclidean topology. The rational sequence topology is given by defining each rational number singleton to be open, and…
Sasha Fomin
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How to show that some set is open?

I know my question will sound stupid, that it should be simple, and I know there are already a lot of questions related to this topic, but I've spent hours on it and I still don't get how to show that a given set is open or not. I'm completely stuck…
justdoit
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Completely regular-Topology

Prove that every metric space is a Tychonoff space. Can somebody please help me to show this space satisfies the completely regular axiom and the $T_1$ axiom.
ccc
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The Unit Sphere $S^{n-1}$ is Path-Connected

I am trying to understand Munkres' proof that $S^{n-1}$ is path-connected. Below is a snippet from the book. It's clear to me that $g$ is surjective; and I concur with him with him when he says it is rather easy to show the continuous image of a…
user193319
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Examples on product topology $ \gg $ box topology?

Suppose that we have $\{X_{\alpha}\}_{\alpha \in J}$, an indexed family of topological spaces. Let $X := \prod_{\alpha \in J}X_{\alpha}$. When we have a map $f_{\alpha} : A \rightarrow X_{\alpha}$ with a topological space $A$. Define $f : A…
user123454321
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How to show that ${\mathbb R}/{\mathbb Z}$ is a compact topological space?

I'm trying to understand the quotient space ${\mathbb R}/{\mathbb Z}$ (with the quotient topology) and I am stuck with the following question: How can I show that ${\mathbb R}/{\mathbb Z}$ is compact? One can either establish a homeomorphism…
user9464
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Vietoris Topology

‎let ‎‎$ X ‎‎‎‎$ ‎be a‎ ‎topological ‎space ‎and‎ ‎$ \operatorname{‎Exp}(X‎)‎‎‎ $ ‎is ‎the set of all ‎closed ‎non-empty subsets of $X$ .‎‎ If $ U , ‎V‎_{‎1‎}‎, V‎_{‎2‎}‎\ldots ‎V_{n}‎$ ‎are ‎the non-empty open subset ‎in ‎$ ‎X‎$‎‎‎,…
M.O
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Single set in a topology

What is meant by a single set in a topological space? The statement goes as: "let $X$ and $X'$ denote a single set in the topologies $\mathcal{T}$ and $\mathcal{T'}$ respectively".
jimm
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Does an open set have an infinite number of points?

Wolfram defines an open set as A set for which every point in the set has a neighborhood lying in the set. Is my understanding correct?: an open set has an infinite number of points because no matter how close you get to the boundary, you still…
stacko
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Cofinite topology on $X \times X$ with $X$ an infinite set

Let $X$ be an infinite set. And consider $(X \times X)_{cof}$ and $X_{cof} \times X_{cof}$. I can see that $(X \times X)_{cof}$ is not finer than $X_{cof} \times X_{cof}$. MY QUESTION: But is $X_{cof} \times X_{cof}$ finer (hence strictly finer)…
CuriousKid7
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a set containing every limit points but not closed

I know that $[0,1]^{\Bbb R}$ with the product topology is not 1st countable. What I want now is to find a subset of $[0,1]^{\Bbb R}$ which is not closed but has all limit points. Does such a set exist? Then, what is it?
jwchoi
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Neighborhood vs open neighborhood

I am a beginner in the concept of general Topology. I have some confusion about the utilization of neighborhood and open neighborhood. $\mathbf{Definition}$: Let $(X,\mathcal{T})$ be a topological space and $x\in X$. $N\subset X$ is called…
Cachorro
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Axioms of closed sets

Let $S$ be a topological space. It means that collection of open subsets of $S$ satisfying the following axioms: $\varnothing$ and $S$ are open any union of open sets is open any finite intersection of open sets is open Besides we may consider…
Aspirin
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