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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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Why are spheres with two holes and cylinders homeomorphic?

I started reading a book on topology and encountered the following in the preliminaries: A sphere with two holes, a cylinder, an annulus, and a disc with one hole are homeomorphic. A sphere with two holes is just an inflated version of a cylinder,…
a_hayler
  • 93
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Product topology with finer/coarser comparison

There is a question in Munkres' Topology which has me a little confused: Let X have topologies $\mathfrak{T}$, $\mathfrak{T'}$, and Y have a topologies $\mathfrak{U}$,$\mathfrak{U'}$. Show that if $\mathfrak{T} \subset \mathfrak{T'}$ and…
user68193
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What does it mean to equip a subset with the subspace topology?

This goes for equipping any set with any topology. Suppose that I want to equip a subset $A \subseteq X$ where $X$ is a topological space, with the subspace topology. I know that by definition of the subspace topology that $\tau_A = \{U \cap A | U…
Joey
  • 904
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covering a space with closures of disjoint sets from a basis

We are given a compact metric topological space $X$ and a base $\beta$ of its topology. Is it always possible to find a subset of the basis, say $\beta_0 \subset \beta$, such that each point of the space is contained in a closure of a set from…
Jakub Konieczny
  • 12,588
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Confirmation of proof that homeomorphism preserves dimension

It seems that I am being able to show that the dimension of a homeomorphic image of a topological space is equal to the dimension of the original space itself. In what follows, I define the dimension of a topological space $X$ in accordance with…
AK12N1
  • 157
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Confusion about Lemma 13.2 in Munkres' topology (property which implies that a collection is a basis for a topology)

Lemma 13.2 and its proof confuse me. $X$ is a topological space and $C$ is a collection of open sets of $X$ satisfying a property. A specific topology is not mentioned in the lemma. In the proof, he assumes the properties of topologies are…
fail
  • 53
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If $Z\subset Y\subset X$ and $Z$ is closed in $Y$, then $Z=Y\cap\overline{Z}$

Suppose $X,Y,Z$ are topological spaces such that $Z\subset Y\subset X$ and $Z$ is closed in $Y$. Is it then true that $Z=Y\cap\overline{Z}$? (Here $\overline{Z}$ is the closure of $Z$ in $X$). If $Z$ is closed in $Y$, then $Z=C\cap Y$ for some set…
ponchan
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Can we define a topology on $\mathbb{R}^2$ using lines instead of circles or squares?

Does it make sense to define a topology on the coordinate grid $\mathbb{R}^2$ using the horizontal and vertical lines as "building-blocks"? $B_{1, y} = \{ (x,y): y \in \mathbb{R} \}$ vertical lines are closed $B_{2, x} = \{ (x,y): x \in \mathbb{R}…
cactus314
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$X$ topological space. $A$ open $A \cap Y = \emptyset \ \ \Longrightarrow A \cap \bar{Y} = \emptyset$?

I know this is an easy question, but I cannot demonstrate it properly. Suppose by contradiction that $A \cap \bar{Y} \neq \emptyset$. Then $\exists \ x \in A \cap \bar{Y}$. I need help formalizing this reasoning (or correcting if it is wrong) By…
Riccardo
  • 7,401
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Closed set implies non-open set?

Given $X$ a metric space and $E$ is a strict subset of $X$ that's non-empty and is closed in $X$, is it true that $E$ is not open? My guess is no, considering I can form the metric space $X=(-1,1) \cup 2$ and then $E = (-1,1)$. Every limit point of…
DanZimm
  • 5,781
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$\bigcap\limits_{\alpha\in I}X_\alpha$ is homeomorphic to a closed subspace of $\prod\limits_{\alpha\in I}X_\alpha$

Let $Y$ is a topological space and for $\alpha\in I$, $X_\alpha\subset Y$. Then $\bigcap\limits_{\alpha\in I}X_\alpha$ is homeomorphic to a closed subspace of $\prod\limits_{\alpha\in I}X_\alpha$. My main problem in proving this proposition is…
TXC
  • 1,332
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Can this ``almost injective'' function exist?

Let $\pi: X\to Y$ be a surjective function between the compact, metric and connected spaces $X$, $Y$, and $Y_0 = \{y\in Y: \#\pi^{-1}(y)>1\}$. Suppose that: $Y_0$ is dense in $Y$, $Y\setminus Y_0$ is a dense $G_\delta$ in $Y$, and $\#\pi^{-1}(y)…
RB1995
  • 319
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Idempotence of the interior of the closure

I'm reading the Complex Analysis text by Ahlfors. I'm stuck on exercise 5 on chapter 3: Prove that $\overline{\overline{\overline{{\overline{X}}^c}^c}^c}^c=\overline{{\overline{X}}^c}^c$. I've manged to rephrase the question as $…
user1337
  • 24,381
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Question regarding connected subspaces

Let $X$ be a topological space and let $A,B\subseteq X$ be closed in $X$ such that $A\cap B$ and $A\cup B$ are connected (in subspace topology) show that $A,B$ are connected (in subspace topology). I would appreciate a hint towards the solution :)
Serpahimz
  • 3,781
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2 answers

The interior of open set in a convex set is not empty

Definition The upper half-space $H^n$ in $\Bbb R^n$ is the set of those $x\in\Bbb R^n$ such that $x_n\ge 0$. So I ask if it is true that any not empty and open set $U$ in $H^n$ has interior (in $\Bbb R^n$) not empty. Probably this is a consequence…
Antonio Maria Di Mauro
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