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
8
votes
1 answer

a example of two space that are homotopy equivalent but not homeomorphic

Is $D^2$ and the point space $P$ containing a point of $D^2$ homeomorphic? Are the two space of same homotopy type? I am seeking for a example of two space that are homotopy equivalent but not homeomorphic.
user151456
  • 155
8
votes
1 answer

Cantor set's topology

Let $C\subset [0,1]$ be the usual Cantor set. Is $[0,a]\cap C$ both open and closed in the relative topology of $C$, whenever $a>0$? Of course it is closed, so the question is about being open.
SteelplantFailure
  • 95
8
votes
1 answer

What is the topology generated by a basis?

Reading Munkres' text on Topology, we get the fairly straight-forward definition of a basis: $\mathcal{B}$ is a basis for a topology on $X$ if $\mathcal{B}$ is a collection of subsets of $X$ such that (1) For each $x\in X$, there is at least one…
Andrew Thompson
  • 4,431
8
votes
2 answers

Algorithms to find simple closed curves on a surface

A closed curve on a surface can be represented by the boundary it intersects. Suppose the original surface is created by glueing $a$ to $A$, $b$ to $B$ etc. Pick an orientation, each time it intersects a boundary, add the boundary to the…
Chao Xu
  • 5,768
8
votes
1 answer

Proof of Pasting Lemma

I am trying to understand the proof of the Pasting Lemma. I have found several proofs but I am missing something from all of them. Wikipedia has: Statement: Let $X,Y$ be both closed (or both open) subsets of a topological space $A$ such that $A = X…
mb7744
  • 1,292
8
votes
1 answer

A Polish space which is not locally compact

I want to find an example of Polish space which is not locally compact. I am thinking about the space of all continuous function from $[0,1]$ to $R$, endowed with the metric $d(f,g) = \sup_{x\in [0,1]}|f(x)-g(x)|$. I know this space is complete.…
Petite Etincelle
  • 14,767
8
votes
1 answer

Why is the lower limit topology not metrizable?

I tried proving it from the second countable axiom, but couldn't figure out how.
user148728
  • 161
8
votes
1 answer

How do you visualize real projective $n$-space?

What I'm asking for is how you visualize this. Definition. Define an equivalence relation on $S^n$ by $x \sim \pm x$ for all $x \in S^n$. Then the quotient space $S^n/ \sim$ is called the (real) projective $n$-space and is denoted $P^n$. Lecturer…
simplicity
  • 3,694
8
votes
1 answer

How many sets can you get by taking closures, complements, and intersections?

The Kuratowski closure-complement problem yields 14 sets which can be formed by taking the closure and the complement of a single set. But if I want to also include such sets as the frontier or boundary, I also need to be able to take intersections…
Mario Carneiro
  • 27,438
8
votes
2 answers

half space is not homeomorphic to euclidean space

We define, half space $H^n$ = $\{(x_1,x_2,...,x_n) | x_n \geq 0\}$. Can anyone suggest, how to prove that $H^n$ is not homeomorphic to the euclidean space $R^n$.
user93620
  • 313
  • 2
  • 5
8
votes
0 answers

When is a free and transitive $G$-space homeomorphic to $G$? And when is it (not) a torsor?

In response to comments: let $G$ be a topological group. A $G$-space is a topological space $X$ equipped with a continuous right group action: a continuous function $X \times G \to X$, $(x, s) \mapsto xs$, such that $x(st) = (xs)t$ and $x1 =…
Aleksei Averchenko
  • 8,023
8
votes
3 answers

Möbius strip with edge identified - constructing map?

I'm trying to show that the Möbius strip with boundary circle identified to a point is homeomorphic to $P^2$ (real projective space). I get geometrically why this is so, but, how does one generally construct maps between these spaces to show that…
Joann
8
votes
2 answers

Quotient Topology

Lets $X$ is a topological space and $Y$ is some subset of $X$. How we define topology of quotient space $$X/Y=\{x\in X~|~x\sim y\Leftrightarrow x, y\in Y\}.$$ Thanks.
Aspirin
  • 5,659
8
votes
0 answers

"Implicit" subcover

I'm learning about compactness in topological spaces and I wonder if there is an example of a compact topological space $X$ and an open cover $\{U_\alpha\}$ of $X$ for which we can't show explictly a finite subcover. I thought about it for a while…
user119194
  • 81
  • 1
8
votes
2 answers

From Counterexamples in Topology, question on the lower limit topology

On page 75 of Counterexamples in Topology, the author writes that the lower limit topology on $\mathbb{R}$ is separable since $\mathbb{Q}$ is dense in $\mathbb{R}$. Could someone offer more detail on why this is so? I know…
Gotye
  • 545
1 2 3
…
99 100