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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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Interior of closure of an open set

The question is is the interior of closure of an open set equal the interior of the set? That is, is this true: $(\overline{E})^\circ=E^\circ$ ($E$ open) Thanks.
Ian
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Do we need nets to be indexed by directed sets?

I recently had a look into general topology and now I am trying to wrap my head around the notion of a net. I understand its definition as a map from an upward directed set into our topological space, but I do not get, why it has to be this general.…
Liandarin
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How to show closed subsets of $\mathbb{R}$ are $\sigma$-compact?

How to see that every uncountable closed subset $A$ of $\mathbb{R}$ is the union of countably many compact subsets of $\mathbb{R}$? Thanks ahead.
user65914
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Doubt about proof in Tube Lemma

So i have been studying topology and when proving that the finite product of compact spaces is going to be compact we have to use the tube Lemma, and we have to prove it. I have a question about the proof : Well we start by covering $ x \times Y $…
Someone
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Arithmetic progression topology

I'm being asked to prove that $\mathcal{A}=\left\lbrace A_{a,b}:a,b\in\mathbb{Z}\right\rbrace$ is a basis for a topology on $\mathbb{Z}$, being: $$A_{a,b}=\left\lbrace a+nb:n\in\mathbb{Z}\right\rbrace=\left\lbrace…
MyUserIsThis
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Equivalence relation on a topological space

Define a relation on a topological space $X$ by $x \sim y$ iff there is a connected subspace $A \subseteq X$ that contains both $x$ and $y$. Prove this is an equivalence relation on $X$. I figured out reflexivity and transitivity. The one I am…
Roger
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Cofinite —> Discrete adding infinite subsets

what is the min number of infinite sized subsets you need to add to a cofinite topology (making it no longer cofinite) to yield the discrete topology given the underlying set is countably infinite?
DeeDee
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A compact subset of $\mathbb{R}^\infty$ must be contained in some $\mathbb{R}^n$

I'd like to prove that a compact subset of $\mathbb{R}^\infty$ must be contained in some finite dimensional space $\mathbb{R}^n$. Here, $\mathbb{R}^\infty$ is the set of all eventually zero sequences of reals, and the topology on $\mathbb{R}^\infty$…
user500144
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How to list all possible topologies

I am begginer and have no more idea about listing all the topologies on a given set. Precisely, given a finite set $X=\{a,b,c\}$ kindly guide me to list all its possible topologies.
Mathematician
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Does Every Subset of a Separable Topological Space have Countably Many Isolated Points?

This is almost certainly a duplicate, but I keep seeing this result on metric spaces, not topological ones. Let $(X,\tau)$ be a topology. A set $A\subset X$ is dense if $A\cap B\neq\emptyset$ for all $B\in\tau$. We say $(X,\tau)$ is separable if…
P-addict
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$f,g : X \rightarrow Y$ are continuous functions and Y is an ordered set then $\{x \in X: f(x) \leq g(x)\}$ is closed

$f,g : X \rightarrow Y$ are continuous functions and Y is an ordered set then $\{x \in X: f(x) \leq g(x)\}$ is closed I saw a proof of this by showing that its complement is open. But the way i started thinking about the problem was trying to prove…
Someone
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About the complement of a compact set.

I tried to prove that If $\omega$ is the complement of a compact set, then $\omega$ has only one unbounded component. I know that the complement of a large disc containing the compact set is unbounded and connected. I see the prove saying that if…
John He
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Confusion on locally compact Hausdorff

I am confused on the following theorem. Let X be a space. Then X is locally compact Hausdorff if and only if there exists a space Y satisfying the following conditions: (1) X is a subspace of Y (2) The set $Y-X$ consists of a single point. (3) Y is…
user404735
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Interior of arbitrary products

$\newcommand{\Int}{\operatorname{Int}}$I need help to show that in the product space $R^\omega$, $\Int((0,1)^\omega)=\emptyset$ Hence $$\Int\left(\prod A_ {\alpha}\right)\neq \prod \Int\left( A_ {\alpha}\right)$$ does not hold true, where $\alpha…
Klara
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Topology generated by the subcollections of compact sets of a metric space

For a metric space $(X,d)$ let $\mathbb{K}X$ be the collection of compact subsets of $X$. Give $\mathbb{K}X$ the topology generated by the sets: $$W(U,K)=\{C\in\mathbb{K}X:(K\cup C)-(K\cap C)\subseteq U\}$$ Here $K\subseteq X$ is compact,…
Kato yu
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