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
4
votes
4 answers

Prove that an open interval and a closed interval are not homeomorphic

Prove that an open interval $(a,b)$ and a closed interval $[c,d]$ are not homeomorphic. I'm trying to prove this statement but have only vague ideas on how to start. How may I use the property of connectedness to show this?
nomadicmathematician
  • 5,511
4
votes
2 answers

Is a basis for a given topology always closed under finite intersections?

Define a basis $S$ for a given topology $\delta$ on $X$ as a subset of $\mathcal{P}(X)$ which satisfies the following conditions: $S \subseteq \delta$ and, for every $U \in \delta$ and every $p \in U$, there is a $V \in S$ such that $p \in V$ and $V…
Nagase
  • 5,467
4
votes
2 answers

first countable $\Leftrightarrow$ compact and Hausdorff?

Can someone give me a short sketch of a proof or a space that serves as a counterexample to the fact that every first countable space is characterized by being compact and Hausdorff (or, stronger than Hausdorff, metrizable) ?
resu
  • 1,782
4
votes
1 answer

Topological Structure of Finite Set

I encounter with a problem in Topological Manifold written by Lee: How many different topological structure of $\{1,2,3\}$? It is easy to make a list of the question, and the answer is $9$. However I am interested in the more general case:…
gaoxinge
  • 4,434
4
votes
1 answer

Is there an injective continuous map $\mathbb{R}^2 \rightarrow \mathbb{R}$?

It is commonly known fact that there exists a continuous surjective map $\mathbb{R} \rightarrow \mathbb{R}^2$. So it bids to ask: Is there an injective continuous map $\mathbb{R}^2 \rightarrow \mathbb{R}$?
bbxlmnistvii
  • 177
  • 5
4
votes
1 answer

Can euclidean space be written as $X \times X$ for some topological space $X$?

Can $\mathbb{R}^n$ be written as $X \times X$ for some topological space $X$? This is obviously true if $n$ is even. Take $X = \mathbb{R}^{n/2}$. However, I'm unsure about $n$ odd.
user141380
  • 41
4
votes
1 answer

Proving that $f(\bar Z)\subset\overline {f(Z)}$ when $f$ is a continuous map

I'm trying to solve this question from my textbook: Let $f:X\rightarrow Y$ be a continuous map and let $Z \subset X$. Prove the inclusion $f(\bar Z)\subset\overline {f(Z)}$. Thanks in advance for any help!
Serph
  • 165
4
votes
1 answer

Topological terminology: neither open nor closed

In topology, a set which is both open and closed is often called "clopen." Is there a commonly used term for sets which are neither open nor closed?
MikeC
  • 1,473
4
votes
3 answers

I need someone to explain this proof from James Munkres' Topology.

The author writes $(X-C)\cap Y = Y-A,$ and, also, $A=Y \cap (X-U)$. I was wondering how is that something anyone writing an original proof of the theorem saw and if there is an analytic proof that the equalities holds true. Thanks for your help.
Noor Aslam
  • 255
4
votes
1 answer

$g$ a restriction of a homeomorphic function $f$, $g$ also homeomorphic?

Let $f:X \to Y$ be a homeomorphism between topological spaces $X$ and $Y$. Let $A \subseteq X$ and $B = f(A) \subseteq Y$ (given the subspace topology) and let $g: A \to B$ be the restriction of $f$ given by $g(a) = f(a)$ for $a \in A$. Showing that…
sonicboom
  • 9,921
  • 13
  • 50
  • 84
4
votes
0 answers

Is the functor associating a bundle with a structure group to a principal bundle faithful?

Consider a (Cartan) principal G-bundle $\xi: X \to B$, and a left $G$-space $F$. We construct the bundle $\xi[F]: X_F \to B$ associated with $\xi$ with a fiber $F$ as usual. Now for each morphism $(u, f)$ between principal bundles $\xi: X \to A$ and…
Aleksei Averchenko
  • 8,023
4
votes
1 answer

Existence of Homeomorphism?

Is there any homeomorphism between $(X,T^1)$ and $(X,T^2)$ where $T^1$ and $T^2$ are topologies on X such that $T^1$ is a proper subset of $T^2$.
Abcd J
  • 1,140
4
votes
0 answers

Connected subsets of the closed unit disc

I have been trying to solve the following problem : Does there exist two disjoint connected subsets of the closed unit disc in $\mathbb R^2$ such that one contains the points $(1,0)$ and $(-1,0)$ and the other contains the points $(0,1)$ and…
user137806
  • 41
4
votes
1 answer

Metric space in Topology class

On the set of integers $\mathbb Z$, show that the function d, defined as follow, is a metric : $$ d(x,y) = \begin{cases} 0 & \text{if } x=y \\ \min\{1/n! \mid n! \text{ divides } |x-y|\} & \text{if } x\neq y \end{cases}$$ metric space…
jakeoung
  • 1,261
4
votes
1 answer

Why topological embedding continuous?

Problem Why do we require a topological embedding to be continuous? Compared to other categories we want a space to be isomorphic to some subspace. Translating this to topological spaces that is saying there is a homeomorphism to a subset in the…
C-star-W-star
  • 16,275
1 2 3
…
99 100