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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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(Un-)Countable union of open sets

Let $A_i$ be open subsets of $\Omega$. Then $A_0 \cap A_1$ and $A_0 \cup A_1$ are open sets as well. Thereby follows, that also $\bigcap_{i=1}^N A_i$ and $\bigcup_{i=1}^N A_i$ are open sets. My question is, does thereby follow that $\bigcap_{i \in…
Haatschii
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Do there exist general conditions under which we can conclude that continuity on a topological space is detected by $\mathbb{R}$?

Whenever $X$ is a topological space, let us say that continuity on $X$ is detected by $\mathbb{R}$ iff for all functions $f : X \rightarrow Y$ where $Y$ is another topological space, we have that if every continuous function $c : \mathbb{R}…
goblin GONE
  • 67,744
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3 answers

Smallest open set in $\mathbb{R}$ that is supset of $\mathbb{Q}$

My problem is that intuitively I would think the whole $\mathbb{R}$ is the only open supset of $\mathbb{Q}$. However this is not true since I can take out for example $\pi$ an I still have an open subset. Now I have two questions, first how to…
Haatschii
  • 1,007
9
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4 answers

Is it true that if two open sets in a topological space intersect each other and share the same boundary, they are the same?

Suppose we have a topological space $(X,\mathcal T)$. Open sets $A,B$ have the following properties: $A\cap B\ne\emptyset$ $\partial A=\partial B$ Then $A=B$. Is it correct? If so, how to prove it?
Drake Marquis
  • 1,843
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Complement of a bounded set $B$ in $\mathbb{R}^{n}$ has exactly one unbounded component.

I'm working on a problem that I think has a very intuitive result, but I'm having a hard time coming up with a rigorous proof. The problem reads If $B$ is a bounded subset of $\mathbb{R}^{n}$ where $n\geq2$, then the complement of $B$ in…
David K.
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Union of topologies

I need to prove that if $\tau_1$ and $\tau_2$ are topologies of a set $X$, then $\tau_1 \cup \tau_2$ is not necessarily a topology on $X$. I'm looking for counterexamples. I have one: consider $X=\{a,b,c\}$, and $\tau_1=\{\emptyset, X, \{a,c\}\}$…
Hiperion
  • 1,763
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A challenging question about T1 spaces and countable compactness

I was working through a textbook on topology and I came across a problem I couldn't solve. 1) It is known that if a space is T1, it is countably compact if and only if every countable open cover has a finite subcover. (See below for the definition…
Mark
  • 5,696
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Is a subset of a topological space able to induce a quotient space

As far as I know, a quotient space of a topological space must be defined wrt an equivalence relation on the topological space. But then I wonder what the equivalence relation is in the following example from Wikipedia: In topology, especially…
Tim
  • 47,382
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If a set is compact then it is closed

Show that if a set is compact then it is closed. definitions: Let $A\subset \mathbb{R}$. A point $p\in\mathbb{R}$ is an accumulation point or limit point of $A$ if and only if every open set $G$ containing $p$ contains a point of $A$ different from…
Wolfy
  • 6,495
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What happens if singleton set is not closed

I'm reading Munkres and now is learning the separation axioms. When he starts to discuss regularity and normality, he says "Suppose one-point sets are closed in $X$". Our Prof. also didn't explain much in the class. So I'm quite curious what happens…
Justin Zhou
  • 145
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3 answers

What does compactness in one topology tell us about compactness in another (coarser or finer) topology?

This problem is from section 27 in Munkres' book on topology. Let $\mathcal{T}$ and $\mathcal{T}'$ be two topologies on the set $X$; suppose that $\mathcal{T}'\supset \mathcal{T}$. What does compactness of $X$ under one of these topologies imply…
Nidia
  • 414
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4 answers

Are $\emptyset$ and $X$ closed, open or clopen?

It is indeed a very basic question but I am confused: (1) In an 2013 MSE posting under general topology here, I was told that $\emptyset$ is an open set and therefore I assume $X$ must be open too. (2) But in Wikipedia page on clopen set here, it…
A.Magnus
  • 3,527
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2 answers

Alternative subbasis for the compact-open topology on $C(X,Y)$

Given spaces $X,Y$, where $X$ is Hausdorff, and the topology on $Y$ has basis $\mathcal{U}$, I would like to show that the set $\mathcal{S} := \{ S(K,U) \mid K \subset X, K \text{ compact, } U \in \mathcal{U} \}$, where $S(K,U) = \{ f \in C(X,Y)…
user121410
  • 329
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1 answer

Topological Hangman

Suppose a mysterious adversary has captured me and challenged me to the following topological game. We fix some finite set, say, $X = \{1, 2, 3, \ldots, n\}$, and my adversary secretly constructs a topology $T$ on $X$. My job is to determine $T$ by…
David Zhang
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3 answers

Verify that $x\mapsto (\cos(x),\sin(x))$ from the real line to the unit circle is an open map.

The setup: Let $p:\mathbb{R}\to S^1$ be defined by $x\mapsto (\cos(x),\sin(x))$. Prove that $p$ is open, i.e. $p$ sends open subsets of $\mathbb{R}$ to open subsets of $S^1=\{ (\cos(x),\sin(x))\in \mathbb{R}^2 \ \vert \ x\in \mathbb{R}\}$. What I…
Adrien Vakili
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