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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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how to prove that countably compact D-space is compact

In the paper A survey of D-spaces by Gary Gruenhage it is written that it is easily seen that any countably compact D-space is compact. However I'm not able to show it. Here is my attempt to prove this claim: Let $(X, \tau)$ be a countably compact…
mcihak
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Prob. 3, Sec. 22, in Munkres' TOPOLOGY, 2nd edition: How is this map a quotient map that is neither open nor closed?

Let $\pi_1 \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be projection on the first coordinate. Let $A$ be the subspace of $\mathbb{R} \times \mathbb{R}$ consisting of all points $x \times y$ for which either $x \geq 0$ or $y = 0$ (or both);…
Saaqib Mahmood
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Example 1, Sec. 22, in Munkres' TOPOLOGY, 2nd edition: How to verify that this map is closed?

Let $X$ be the subspace $[0,1] \cup [2,3]$ of $\mathbb{R}$, and let $Y$ be the subspace $[0,2]$ of $\mathbb{R}$. The map $p \colon X \to Y$ defined by $$ p(x) \colon= \begin{cases} x \ &\mbox{ for } \ x \in [0,1], \\ x-1 &\mbox{ for } \ x \in…
Saaqib Mahmood
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Section 22 in Munkres' TOPOLOGY, 2nd edition: How to establish this equivalence?

Let $X$ and $Y$ be topological spaces; let $p \colon X \to Y$ be a surjective map. Then $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is open in $X$ (or equivalently a subset $B$ of $Y$ is…
Saaqib Mahmood
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Let $g: S^2 \to S^2$ be continuous and $g(x) \neq g(-x)\ \forall x$. Prove that $g$ is surjective.

Let $g: S^2 \to S^2$ be continuous and $g(x) \neq g(-x)\ \forall x$. Prove that $g$ is surjective. The hint that if $p \in S^2$, then $S^2 - \{p\}$ is homeomorphic to $\mathbb{R^2}$. It is pretty obvious that one can use the Borsuk-Ulam Theorem,…
Kees Til
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Proving that if a set is both open and closed then it is equal to the real numbers

Prove that if $A$ is both open and closed then $A = \mathbb{R}$ also as one suggested let $A \neq \emptyset$ You may use what ever definition of open and closed you would like, just avoid going into metric spaces, haven't covered that topic yet. My…
Wolfy
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Not open and not closed set

Can somebody provide not open and not closed set? I even cannot imagine what does it mean. Also, I'm having a deal with such problem that there is a bounded countable set in R, and I should provide examples of sets that are: opened; closed; not…
Andrey Andrey
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Spaces in which the closure of every countable subset does not include an uncountable closed discrete subset

What classes of spaces $X$ have the property that that for every countable subset $C \subset X$, $\overline{C}$ does not have an uncountable closed discrete subset? I know every space with countable extent, such as compact spaces, Lindelöf spaces…
Paul
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Request For An Example Of A Continuous Map Relative To The Box Topology On $\mathbb{R}^J$, When $J$ Is Infinite

Let $J$ denote an infinite --- countable or uncountable --- index set. Let $\mathbb{R}^J$ denote the set of all $J$-tuples of real numbers (i.e. the set of all real-valued functions with domain $J$) under the box topology having as basis all…
Saaqib Mahmood
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Characterisation of limit points of subsets of Hausdorff spaces

The theorem which I want to show is the following: For a Hausdorff space $X$ and a subset $A$ of $X$, $x$ is a limit point of $A$ if and only if every neighborhood of $x$ contains infinitely many points of $A$. And this is my answer; (⇒) Suppose…
Darae-Uri
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Connected closed subset in a plane

Could anyone give a hint on the proof of the following fact? Let $X$ be a closed connected subset in a a 2-sphere. Then every connected component of the complement of $X$ is simply connected. It seems to use Jordan curve theorem. If one component…
stephen
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Homework question on whether two quotient spaces are homeomorphic

We have two spaces $X=\{(x,1/n):n\neq 0, n\in\mathbb{Z}, x\in\mathbb{R}\}$ and $Y=\{(x,n):n\neq 0, n\in\mathbb{Z}, x\in\mathbb{R}\}$. On both spaces we introduce the equivalent relation $(x,y)\sim (x',y')$ if $x=x'$ and $y=y'$ or $x=x'=0$. That is,…
Hui Yu
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When is the stabilizer topologically closed?

An exercise in Armstrong says if the topological group $G$ acts on the topological space $X$ by homeomorphisms, then the stabilizer $\text{st}(x)=\{g\in G\mid g(x)=x\}$ is a closed subset of $G$. If $X$ were Hausdorff this would be easy. But he…
Gregory Grant
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A Question on a second countable $T_2$ space

I have a question: Does a second countable $T_2$ space $X$ always have a $G_\delta$-diagonal? (If $X$ is regular, then it is metrizable, and it obviously has a $G_\delta$-diagonal.) Thanks for your help.
Paul
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Prob. 8, Sec. 19 in Munkres' TOPOLOGY, 2nd ed: What is the situation in the box topology?

Let $\mathbb{R}^\omega$ denote the set of all sequences of real numbers, and let $(a_1, a_2, a_3, \ldots ), (b_1, b_2, b_3, \ldots) \in \mathbb{R}^\omega$ be fixed with $a_i > 0$ for all $i= 1, 2, 3, \ldots$. Let the map $h \colon \mathbb{R}^\omega…
Saaqib Mahmood
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