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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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$p$-adic integers $\mathbf{Z}_p$ and $\mathbf{Z}_q$ homeomorphic for $p \neq q$?

Obviously, $\mathbf{Z}_p \not\cong \mathbf{Z}_q$ ($p$-adic integers: $\mathbf{Z}_p = \varprojlim_n\mathbf{Z}/p^n$) for $p \neq q$ as (topological or abstract) groups, but are they homeomorphic as topological (profinite) spaces?
user3267
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Potential Munkres Error? (Impossible!)

This is problem 19.10 (d) from Munkres' topology text, the second edition: Let $A$ be a set; let $\{X_\alpha\}$ be an indexed family of spaces; and let $\{f_\alpha\}$ be an indexed family of functions, $f_\alpha\colon A \to X_\alpha$. Let…
user60499
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About the universal property of initial and final topologies

I have recently seen in my topology course that if $X$ is any set, Given a family of functions $\{f_i : X \to Y_i\}_{i \in I}$, with $(Y_i, \tau_i)_i$ topological spaces, the initial topology on $X$ with respect to this family is the one generated…
qualcuno
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Given continuous $f:\{0,1\}^S\to[0,1]$ with S uncountable and $f(\sigma)=1$, show that $f(\tau)=1$ for some $\tau\neq\sigma$

Let $S$ be an uncountable set and consider the space $\{0,1\}^S$ under the product topology where $\{0,1\}$ is discrete. Let $f:\{0,1\}^S\to[0,1]$ be a continuous function and suppose $f(\sigma)=1$ for some $\sigma\in\{0,1\}^S$. I need to show that…
Anonymous
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Question of whether two given spaces are homeomorphic.

Let $D^2$ be the closed disk on the plane. First we pick an arbitrary point $x\in bd(D^2)$ on $D^2$, and define $X = D^2-\{x\}$. Then define another space $Y$ by removing a homeomorphic image of the closed interval $I$ from the boundary, that is,…
William Sun
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What is meant by pointwise fixed?

I did not find a direct definition so I will give the context : $X$ is compact manifold with boundary. There is no smooth map $f:X\rightarrow \partial X$ that leaves $\partial X$ pointwise fixed. Here what is meant by pointwise fixed? I found that…
mathemather
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What are these two topologies?

I'm a beginner in Topology. Today, this came up my mind: (1) For a set $X$, choose a subset $A\subseteq X$. Let $S\subseteq X$ be a closed set if and only if $(A\subseteq S)\vee (S=\emptyset)$. This is a topology on $X$. (2) For a set $X$, let…
twelvefaces
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distance between two subset , seperation of two subset in a metric space

Given $(X,d)$ a metric space for subsets $A,B$ of $X$, define $$d(A,B)=\inf\{d(a,b):a\in A,b\in B\}$$ could any one confirm me with logic which of the following are/is true and false? if $\bar{A}\cap\bar{B}=\phi$, then $d(A,B)>0$ if $d(A,B)>0$ then…
Myshkin
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If $a\in \mathrm{clo}(S)$, does it follow that there exists a sequence of points in $S$ that converges to $a$?

Let $X$ be a topological space and $S=\{x_n\}$ be a sequence of points in $X$. Suppose $a$ is a point in $X$ such that $a$ is adherent to $S$(that is $a$ is in the closure of $S$),I want to ask if there must exist a sequence $\{y_n\}$ in $S$ such…
abc31415926
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If $X$ is normal and $A$ is a $F_{\sigma}$-set in $X$, then $A$ is normal. How could I prove this theorem?

A topological space $X$ is a normal space if, given any disjoint closed sets $E$ and $F$, there are open neighbourhoods $U$ of $E$ and $V$ of $F$ that are also disjoint. (Or more intuitively, this condition says that $E$ and $F$ can be separated by…
onimoni
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Proof that Sorgenfrey plane is not normal using points x × (-x)

I'm making Exercise 9 of paragraph 31 in Munkres, which is a proof that the Sorgenfrey Plane $\mathbb{R}_l^2$ is not normal. I'm having trouble on part c of the question. The full question is: Let $A$ be the set of all points of $\mathbb{R}_l^2$ of…
user49719
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$f:U\rightarrow \mathbb{R}$ continuous, injective- prove $f(U)$ is open.

If we have that $f:U\rightarrow \mathbb{R}$ where $U$ is open and $U\subset\mathbb{R}$ and $f$ is continuous and injective I want to show that $f(U)$ is open. So I have considered the function restricted to $f:U\rightarrow f(U)$ so that it is a…
hmmmm
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Connected subspaces

I guess there's something wrong with my thoughts about connectedness seen in a subspace of a topological space and I need your help. Let me explain: These are the definitions I have: A topological space $X$ is said to be disconnected if there are…
user55334
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Compact subspaces of the Poset

On page 172, James Munkres' textbook Topology(2ed), there is a theorem about compact subspaces of the real line: Let $X$ be a simply-ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is…
Metta World Peace
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Prove $\mathbb{R} ^n$ is contractible for any $n \geq 1.$

$X$ is contractible if the identity map $i_X : X \to X$ is homotopic to a constant map. Prove $\mathbb{R} ^n$ is contractible for any $n \geq 1.$ Is any contractible space is path connected? I am not sure how to show $\mathbb{R} ^n$ is…
Compact
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