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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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Completely regular space is hereditary

So I look online that the definition of the completely regular if whenever $E\subset X$ is closed and $x\notin E$ there is a continuous function $f:X\to [0,1]$ such that $f(x)=0$, and $f(E)=\{1\}$. Now, on a "fact", they also say they the Tychonoff…
Akaichan
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On existence of subset of $\mathbb{R}^{n}$ homemorphic to $\mathbb{S}^n$

As the title suggest, I have the following question: For which $n\in\mathbb{N}$ does there exists a subset $S\subset\mathbb{R}^n$ so that $S\cong\mathbb{S}^{n}=\{ x\in\mathbb{R}^{n+1}:\|x\|=1 \}$ I have a feeling that the answer should be: For…
miraunpajaro
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Why not just study the consequences of Hausdorff axiom? What do statements like, "The arbitrary union of open sets is open," gain us?

Define that a pair $(X,\tau)$ where $\tau \subseteq \mathcal{P}(X)$, is a Hausdorff space if for all distinct $a,b \in X$ there exist $A,B \in \tau$ such that $$a \in A, \;b \in B, \;A \cap B = \emptyset.$$ Note that a Hausdorff space, according to…
goblin GONE
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Are any two open connected subsets of $\Bbb{R}^n$ homeomorphic?

Are any two open, connected subsets of $\Bbb{R}^n$ homeomorphic? This seems intuitively true.
user63952
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Intersection of Connected Sets

This is an old exam question that I don't have a solution to: Let $X$, a compact Hausdorff (T2) space, and let $\phi$ a family of closed, non-empty, and connected subsets of $X$, such that for every $A, B \in \phi$, $A \subset B$ or $B \subset…
Hila
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Topological way to show , if $X$ finite than there is no bijection to $X\setminus{x}$ while $x\in X$

I got the Idea from this question: <> another thread I got the idea to define a finite set like this: We call a set $A$ finite if the topological space $(A,T)$ is hausdorff iff $T$ is the discrete topology. If my proof isn't wrong this one is…
Dominic Michaelis
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Continuous surjective functions from the unit disk to itself that agree nowhere

Do there exist two continuous surjective functions $f,g:D \to D$ such that $f(z) \neq g(z) $ for all $z \in D$, where $D$ is the closed unit disk?
user62832
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Very Strange Characterisation of Topology

Definition of Topology I don't understand this characterisation of a topology, I don't even comprehend what $g\colon\{ \{\{\}\},\{\{\},\{\{\}\}\}\} \rightarrow T$ should mean? Can somebody explain it to me?
StefanH
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Union of a countable collection of open balls

Show that a subset of $\mathbb{R^n}$ is open if and only if it is the union of a countable collection of open balls. Attempt: I know a set G is open if there exists a $\epsilon >0 $ such that every point $x\in A$ satisfies $||x-y||<\epsilon$.…
Lays
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Quotient space of $\mathbb{R}^2$

This is an example of from the book Topology of Janich. In the picture $X = \mathbb{R}^2$ with standard topology, and the lines represent the equivalence classes, which are closed $1$ dimensional manifolds. The example is aimed to show: Even if the…
Joe_Chao
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Interior, closure and boundary of the sets of all rational and irrational numbers.

Consider $\mathbb Q$, the set of rational numbers, and its complement $\mathbb R\setminus \mathbb Q$, the set of irrational numbers. I noticed that their interiors, closures and boundaries are the same, that is: Interior: $\varnothing$ Closure:…
i_a_n
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A path connected proof in $\mathbb{R}^2$

I'm trying to prove that if $U \in \mathbb{R}^2$ is open and path connected, then for a point $ p \in U$ we have $U \smallsetminus \{p\}$ still path connected. Start by taking $x,y \in U$. As path connected there exists continuous $\gamma : [0,1]…
user53076
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lower limit topology to the metric topology

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function that is continuous from the right; that is, for all $a \in \mathbb{R}$, $\lim_{x \to a^{+}} f(x) = f(a)$ a) Show that $f$ is continuous when viewed as a function from $\mathbb{R}$ with the…
user130306
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A map is continuous if and only if the restrictions are

I want to find out if the following statement is true or false, and prove why: Let $X$ be a topological space. Suppose $X=A \cup B$ and $f:X \rightarrow Y$ is a map whose restrictions to $A$ and $B$ are $f_A:A \rightarrow Y$ and $f_B:B \rightarrow…
user390960
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The general argument to prove a set is closed/open

I am taking a topology course and we are now learning open and closed set. I am a bit confused to how to prove that a set is closed or opened, how should I approach these kind of problems. For example: Question 1: Let $(\mathcal{X},d)$ be an…
Vafa Khalighi
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