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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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Empty space connected? If so, then what are its components?

I encountered that a topological space $X$ is connected if no separation exists. Here a separation is a pair of disjoint non-empty open sets whose union is $X$. Such a separation can only exist if $X$ contains two distinct elements, so $\emptyset$…
drhab
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How to prove that every uniform space is completely regular?

Wikipedia says "every uniform space is completely regular". How to prove that every uniform space is completely regular?
porton
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Are quasicomponents connected in a compact non-Hausdorff space?

Are quasicomponents connected in a compact space? Background: Quasicomponents are connected in a compact Hausdorff space. A non-compact locally compact Hausdorff space may not have connected quasicomponents. It can be shown that a subset is an open…
kaba
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Why is the closure of X equal to X for topological spaces

In the book "Introduction to Topology" by Bert Mendelson, ch3, theorem 4.7 states that for any Topological Space (X, T) $\bar{X} = X$. Now I understand that any subset of $X$ is also a subset of $\bar{X}$, but don't see why $X = \bar{X}$. An answer…
Cedric Martens
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Zero sets in pseudocompact spaces

Let $X$ be a Tychonoff pseudocompact space and $Z$ be a zero-set in $X$ (i.e., there is a continuous function $f:X \to \mathbb{R}$ such that $x\in Z$ if and only if $f(x)=0$). It is known that these conditions imply that $Z$ is a weakly…
Peluso
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Name for topology generated by the standard open sets of reals and every point that isn't zero.

I'm interested in topological spaces that are easy to construct. In particular, I'd like to know what this one is called so I can look it up in pi-base, assuming of course that it's interesting enough to be included there. Let $(\mathbb{R}, \tau^R)$…
Greg Nisbet
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The preimage of a single point is not compact

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous surjection under the ordinary topology. Show that the preimage of a single point, i.e. $f^{-1}(\{t\})$ is not compact. The set is obviously closed, and thus is should be unbounded. I guess we need to…
Isomorphism
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Local compactness exercise

Let $X$ be a Hausdorff locally compact in $x \in X$. Show that for each open nbd $U$ of $x$ there exists an open nbd $V$ of $x$ such that $\overline{V}$ is compact and $\overline{V} \subset U$. My work: Since $X$ is Hausdorff and locally compact…
user10
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When does a maximal sub topology not containing an open set exist?

The motivation is this question. I was quite optimistic in the formalization I provided of the question, it would indeed be the case that such a maximal sub topology exists. Now, I learn that it's less clear than I initially thought (after seeing…
tryst with freedom
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The set of all nilpotent matrices is a closed subset of $M(n,\mathbb R).$

Let $M(n,\mathbb R)$ be endowed with $\|.\|_2.$ Then show that the set of all nilpotent matrices is a closed subset of $M(n,\mathbb R).$ I tried using the continuous map $A\mapsto A^n$ on $M(n,\mathbb R).$ But arbitrary union of closed sets is not…
Sriti Mallick
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Continuous function from $\mathbb{R}^2$ to $\mathbb{S}^2$

Good afternoon, I am currently going through an introductory topology class. An exercise asked us to prove there is no homeomorphism between $\mathbb{R}^2$ and $\mathbb{S}^2$ (and the reason is no continuous bijection from $\mathbb{S}^2$ to…
carfog
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Prove that a compact subset of a Hausdorff space is closed. Give an example which shows that the "Hausdorff" assumption is required.

Prove directly from the definitions, that a compact subset of a Hausdorff space is closed. Give an example which shows that the "Hausdorff" assumption is neccessary. Let $X$ be a topological space and $A \subset X$ compact. Let $x \in X \setminus…
Walker
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Homeomorphism between $\mathbb{R}^{n+1} \setminus \{0\}$ and $S^n\times\mathbb{R}$

How to establish a homeomorphism between $\mathbb{R}^{n+1} \setminus \{0\}$ and $S^n\times\mathbb{R}$? Thanks.
Pedro
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Set is open iff complement is closed in $\mathbb{R^n}$

Is my proof of the following correct? A set $A\subseteq \mathbb{R^n}$ is open iff $\mathbb{R^n}-A$ is closed. A set $A\subseteq \mathbb{R^n}$ is closed iff $\mathbb{R^n}-A$ is open. Proof. Suppose that $A$ is open. We must show that for every…
mrk
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Countable product of finite topological spaces is compact (without Tychonoff's Theorem)

I am trying to prove that a countable product of finite topological spaces is compact without using Tychonoff’s theorem. I tried to prove this through sequences, but I couldn't do it. Also, I don't know if it's possible to do it in this case. How…
Cal22
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