Mathematics Stack Exchange
Mathematics Stack Exchange

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
6
votes
1 answer

Visualizing a Deformation Retraction

It's known that there exists a deformation retraction from the space $\mathbb{R}^3$\ $S^1$ to $S^2 \wedge S^1$, and I thought I had a visualization for it, but now it seems discontinuous. Can anyone help out with describing (or even better…
squiggles
  • 1,903
6
votes
1 answer

Generalization of Urysohn's Lemma

Urysohn's lemma in general topology states: A topological space $X$ is normal (i.e., $T_4$) iff, for each pair of disjoint closed subsets $C, D \subset X$, there is a function $f : X \to [0, 1]$ such that $f(C) = 0$ and $f(D) = 1$. Of course…
Daniel McLaury
  • 24,656
6
votes
1 answer

What is a first countable, limit compact space that is not sequentially compact?

I just read a proof that adds the assumption of T1 to conclude that a first countable, limit compact and T1 space must be sequentially compact, but I didn't understand what happens if we drop the T1 assumption. If $X$ is limit point compact and…
Rodrigo
  • 7,646
6
votes
3 answers

Is the graph of a continuous function from a connected space also connected?

Let X be connected and $f:X \rightarrow Y$ continuous. Is the graph $G$ of this function connected? My thought is yes, which seems pretty intuitively clear. For contradiction, assume $G$ is not connected and $G= U \cup V$. I pick $(x,y)$ in $G$, so…
Jack
  • 255
6
votes
1 answer

Finding intervals not compact, not connected

There is this following situation: For $\mathbb{R}$ we consider the family $\mathcal{S}$ of subsets consisting of all the intervals of type $(m,M)$ with $m
jbuser430
  • 743
6
votes
1 answer

Is the following a Kuratowski closure operator?!

Let $(X,\tau)$ be a cofinite topology, where $X$ is infinite. Then, for every $A\subseteq X$, we define a Kuratowski closure operator with: $$cl(A)=\left\{ \begin{array}{cc} A; & A \; \text{finite}\\ X; & A\; \text{infinite} \end{array} \right.$$ I…
zermelovac
  • 941
6
votes
1 answer

A characterizion of continuity for functions between Hausdorff compacta?

Suppose A and B are Hausdorff compacta and $f:A \to B $ is a function. Note if $f$ is a continuous function then $1.$ $f^{-1}(b)$ is closed in A for each b in B. $2.$ $f(C)$ is compact for each compact subspace $C$ of $A$. Do conditions $1$ and $2$…
Paul Fabel
  • 629
6
votes
1 answer

Taking the connected sum of four closed disks... What do we get?

If we take the connected sum of four closed disks $S = 4 \mathbb{\overline{D}} = \mathbb{\overline{D}} \# \mathbb{\overline{D}} \# \mathbb{\overline{D}} \# \mathbb{\overline{D}}$, what does $S$ look like and how do we describe the boundary? (This…
St Vincent
  • 3,070
6
votes
4 answers

Topology $\text{i})$ What is a topology? $\text{ii})$ What does a topology induced by a metric mean?

I am now trying to understand what a topology and a topological space is. Yes, I know the "formal" or "mathematical definition" of it, it is in my notes so it's easy for me to reiterate that. Please bear with me as I am trying my best to express my…
Melba1993
  • 1,111
6
votes
3 answers

Prove that open half planes are open sets

An open half plane is a subset of $\Bbb{R}^2$ in the form $\{(x, y)\in \Bbb{R}^2\vert \space Ax + By
Taylor B
  • 61
6
votes
2 answers

'Complementary topology'?

Let $(X,\mathscr{T})$ be a topological space, and define a new topology $\mathscr{T}'$ on $X$, such that the collection of closed sets $\mathscr{C}$ of the topology $\mathscr{T}$ form a sub-basis for $\mathscr{T}'$. Lets refer to $\mathscr{T}'$ as…
snulty
  • 4,355
6
votes
3 answers

Is a closed subset of isolated points in a compact set necessarily finite?

If I have a compact set $A$ and a closed subset $\Sigma \subset A$ which only contains isolated points (that is, none of them is a limit point). Does the compactness of $A$ then force $\Sigma$ to have finite cardinality ? Here is my attempt at a…
harlekin
  • 8,740
6
votes
1 answer

How to show that $x/|x|$ is continuous?

This is from p.156 of Topology, Munkres: The unit sphere $S^{n-1}$ in $\mathbb{R}^n$ is path-connected, since it is the continuous image of the surjective function $g: \mathbb{R}^n -0\to S^{n-1}$ by $g(x)=x/|x|$. (Note that the punctured euclidean…
Gobi
  • 7,458
6
votes
1 answer

Is a locally countable family of open subsets of an separable space countable?

Definition A topological space $(X, \mathfrak{T} )$ is said to be an separable space if $X$ contains a contable subset $D \subseteq X$ such that $D$ is dense in $X$ Definition Let $(X, \mathfrak{T} )$ be a topological space and $\mathfrak{F}$ be a…
Peter Hu
  • 61
6
votes
1 answer

What type of surface is it?

The picture shows Sphere, Torus, Klein Bottle and Projective Plane, respectively: What about the following one? Is it also Projective Plane? : PS inside triangles and color in shapes are irrelevant.
user231343
1 2 3
…
99 100