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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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ANR is locally contractible

Recall that a space $X$ is contractible if there exists a homotopy $h:X\times [0,1]\to X$ such that $h$ is equal to the identity map on $X\times\{0\}$ and $h$ is constant on $X\times\{1\}$. Please help me out since I'm stuck with this question,…
C_M
  • 711
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2 answers

Point in which all three sets meet

The entire 2-dimensional plane is covered by 3 sets: Blue, Green and Red. It is given that: All sets are closed. All sets are interior-disjoint (but may meet at their boundaries). Blue is bounded. Blue meets both Red and Green (i.e. their…
Erel Segal-Halevi
  • 10,555
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2 answers

$\mathbb {R}^{\omega}$ in the box topology is not first countable.

I'm trying to show that $\mathbb {R}^{\omega}$ in the box topology is not first countable. But I cannot come up with a contradiction by assuming that for each $x \in \mathbb {R}^{\omega}$, there is a countable basis. My intuition is that I need to…
nomadicmathematician
  • 5,511
6
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Cluster point of a sequence and limit point of some subsequence

In a topological space $X$, quoted from Wikipedia: A point $x ∈ X$ is a cluster point of a sequence $(x_n)_{n ∈ N}$ if, for every neighbourhood $V$ of $x$, there are infinitely many natural numbers $n$ such that $x_n ∈ V$. If the space is…
Tim
  • 47,382
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1 answer

If $C_b(X)$ separable then $X$ is a compact metrizable space

I am very familiar with the proof of the following statement: If $X$ is a compact Hausdorff space such that the Banach algebra $C(X)$ is separable, then $X$ is metrizable. Can this be used to prove a more generalized version of this statement with…
josh
  • 4,041
6
votes
3 answers

Gluing a Möbius strip into a sphere.

If we take a square and identify opposite sides, we get a torus. If we change the direction of one pair of sides we get the Klein bottle. If we change the direction of both sides, we get first a Möbius strip, then when I tried to glue the opposite…
Laylady
  • 273
6
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2 answers

boundary of a subset of a subspace of a given topological space

Suppose $T$ a topological space and $S\subseteq T$ a subspace equipped with the subspace topology inherited from $T$. Take a subset $H\subseteq S$. I'd like to prove that $\partial_S H=S\cap \partial_T H$ (where $\partial_X A$ denotes the boundary…
fatoddsun
  • 2,169
5
votes
3 answers

Understanding the Co-finite Topology on R

I'm looking to gain a better understanding of how the cofinite topology applies to R. I know the definition for this topology but I'm specifically looking to find some properties such as the closure, interior, set of limit points, or the boundary…
John MatthewS
  • 167
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Can open sets in different dimensions be homeomorphic to each other?

Assume $U$ is open in $\mathbb{R}^m$ and $V$ open in $\mathbb{R}^n$, $U\cong V$. Does it imply $m=n$?
Fan
  • 1,115
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How to prove $D^2\setminus\{0\}$ is not homeomorphic to $\mathbb{R}^2\setminus\{0\}$?

Here $D^2$ denotes the closed unit disk in $\mathbb{R}^2$. I know that $D^2$ is not homeomorphic to $\mathbb{R}^2$ as $D^2$ is compact. Intuitively I believe that $D^2\setminus\{0\}$ is not homeomorphic to $\mathbb{R}^2\setminus\{0\}$. However, when…
NECing
  • 4,095
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1 answer

Two proofs of cartesian product in topological space

Let $(X \times Y, \tau)$ be cartesian product of topological spaces $(X, \tau_X)$, $(Y, \tau_Y)$. Let $ A \subset X$, $ B \subset Y$. A) Prove that $\overline{A\times B}= \overline{A} \times \overline{B}$ B) Prove that $(A \times B)^d = (A^d \times…
user184414
  • 125
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A few questions on the properties of $\mathbb{R} ^ {[0,1]}$

Given the topological space $X=\mathbb{R}^{[0,1]}$ with the product topology, there are several properties regarding to $X$ which I am not sure if are true/false. Is $X$ metrizable? I'm having trouble on how I can prove/disprove this, and I'm not…
Zhan I.s.
  • 233
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(Topological Groups) Show that $g_{\alpha}(x) = x *\alpha$ is a homeomorphism of G.

Note: This is not homework. Can someone please verify my proof or offer suggestions for improvement? Let $\alpha$ be an element of a topological group $G$. Show that the map $g_{\alpha}: G \longrightarrow G$ defined by $$g_{\alpha}(x) = x *\alpha$$…
user154185
  • 2,096
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1 answer

How to draw $S^1\times I$

I always thought $P=S^1\times I$, where $S^1$ is the circle and $I=(0,1)$ with the standard topology is the surface of the cylinder, but I was reading a book which says me another thing: Even, if the book is right, I didn't understand why $P$ is…
user42912
  • 23,582
5
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2 answers

A space $X$ is locally connected if and only if every component of every open set of $X$ is open?

It is claimed on the wikipedia page that a space $X$ is locally connected if and only if every component of every open set of $X$ is open without any proof. What is the proof behind this fact? Am I correct in assuming this in turn implies that a…
Jeffrey Yellow
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