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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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Stone-Cech compactification of discrete space is totally disconnected

Let $X$ be a discrete space; consider the space $\beta(X)$. a) Show that if $A\subset X$, then $\overline{A}$ and $\overline{X-A}$ are disjoint, where the closures are taken in $\beta(X)$. b) Show that if $U$ is open in $\beta(X)$, then…
RFZ
  • 16,814
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2 answers

How can I show that this space is regular

Consider the collection of all sets of the form $U \cup (V\cap \Bbb{Q})$ where $U,V$ are open in standard topology on $\Bbb{R}$. I have shown that such a collection is a topology which I call $\mathcal{T}$. Now I can show that $\mathcal{T}$ is…
user23086
6
votes
1 answer

Can every two points be connected by a simple curve?

Let $X$ be a path-connected subset of $\mathbb{R}^2$. For $x,y\in X$, $x\neq y$, does there necessarily exist a simple curve connecting $x,y$? In other words, is there an injective continuous map $\gamma:[0,1]\to X$ such that $\gamma(0)=x$,…
Yuxiao Xie
  • 8,536
6
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1 answer

Topological classification of a finite union of open balls in $\mathbb{R}^n$

What are the possible topological shapes (i.e. up to homeomorphism) of a finite union of open balls in $\mathbb{R}^n?$ For example, for $n = 1$, open balls are just open intervals and a finite union of open intervals is just a disjoint union of open…
CNS709
  • 1,657
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Quotient space definition

I am trying to solve a homework problem where a quotient space is defined in a way that I do not understand, so I will ask a more simplified question. I understand that $$ [0,1]/0 \sim 1, $$ means that you sew the points 0 and 1 together, getting…
Mr. Fegur
  • 862
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1 answer

Uspenskij-Tkachenko Theorem

Theorem: For a Moscow space $X$, every $G_\delta$-dense subset $Y$ of $X$ is C-embedded in $X$. Proof $^{1}$: Assume that $Y$ is not C-embedded in $X$. Then, as it is easy to see, there are open subsets $V_1$ and $V_2$ of $Y$ such that their…
M.Sina
  • 1,690
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Recognizing an adjunction space

I am currently studying Adjunction spaces using Brown's Topology and Groupoids. I am having trouble understanding exercise 4.5.3: Let $B$ be a closed subspace of $Q$. For each $λ=1,\dots,n$, let $f_λ:X_λ→Q$ be a map, and let $A_λ$ be a closed…
bd99
  • 121
6
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Show that the subset $D= \{(x,y)~|~x \ne 0, y \ne 0\}$ of the plane is open

$D= \{(x,y)~|~x \ne 0, y \ne 0\}$ I'm thinking I can show that the set is open for $x \gt 0$ and $y \gt 0$ using disks. Maybe I could do the same for $x \lt 0$ and $y \lt 0$. But this process seems too long.
iuppiter
  • 395
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2 answers

A locally metrizable space

Is every locally metrizable space always first countable? A locally metrizable space $X$ means for every point $x\in X$ has an open nbhd such that it is metrizable. Thanks for help.
user65914
6
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2 answers

Help understanding how a topological cone is constructed.

This is the definition of the cone from wikipedia: I have trouble understanding why the cone has that particular shape. Based on what I understand about equivalence relations and quotients, shouldn't the cone consist of copies of X over the length…
ensbana
  • 2,277
6
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2 answers

Collection of inverse images of a topology under a function $f$

I want to prove the following: Let $(Y,\tau)$ be a topological space and $X\neq \emptyset$ a set. Let $f$ be a function from $X$ to $Y$. Then, $\tau_1=\{f^{-1}(S): S\in \tau\}$ is a topology on $X$ My attempt: $\emptyset \in \tau$ and $Y\in…
mrk
  • 3,075
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Let $X$ be connected and locally connected. Let $f:X\to [0,1]$ be bijective continuous function, then $X$ is homeomorphic to $[0,1]$

Let $X$ be connected and locally connected. Let $f:X\to [0,1]$ be continuous and bijective function, then $X$, $[0,1]$ are homeomorphic. My attempt Let $A\subset X$ be non-empty open set, let's see f(A) is open. we take $y\in f(A)$, then there…
John Ederson
  • 61
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4 answers

Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$

Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus \{(0,0)\} $.
user9080
  • 69
6
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1 answer

Why does the support must be closed?

Apparently it is important that the support is defined as the closure of $\{f \neq 0\}$. Because of that condition globalization is allowed as the exercise below indicates. However, I have no idea why it is so important that the support is defined…
LK4
  • 127
6
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3 answers

Prove that $U$ is an open set of a topological space $X$ iff for all $A\subset X$ we have $Cl(U\cap Cl(A))=Cl(U\cap A)$

I'm stuck with this one, is the 14th exercise on my notes and the most difficult so far. Prove that $U$ an open set of a topological space $X$ iff for all $A\subset X$ we have $Cl(U\cap Cl(A))=Cl(U\cap A)$. I've been struggling thinking both…
Relure
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