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
5
votes
2 answers

Product topology is discrete

Willard's General Topology says: For each $\alpha\in A$, let $X_{\alpha}$ be discrete topological space. Then $\prod_{\alpha\in A}X_{\alpha}$(under product topology) will be a discrete space if and only if A is finite. But, if $X_{\alpha}=\{1\}$…
Silent
  • 6,520
5
votes
1 answer

Intersection of Simply-Connected Sets

Let $U,V$ be two simply connected subsets of a topological space. Prove or disprove: $U \cap V$ is simply connected.
pre-kidney
  • 30,223
5
votes
2 answers

Is R with $j_d$ topology totally disconnected?

Let consider the topological space $R_j=(\mathbb{R},j_r)$ where $j_r$ is generated by right side open intervals, i.e $[a,b)$ for $a,b \in \mathbb{R}$; note that this topology includes the euclidean topology. $R_j$ is not a connected space, because…
Temitope.A
  • 1,710
5
votes
1 answer

Every set in $T1$-space is saturated.

Let $(X, \mathcal T)$ be a $T_1$-space. Now for every $x \in X$, we know that $\{x \}$ is closed in $X$, that is $X \setminus \{x\}$ is open. Consider any subset $S \subseteq X$. We have that $\displaystyle S = \bigcup_{x \in S} \{x\}$ is closed,…
user290425
  • 802
5
votes
0 answers

$X$ and $Y$ are homeomorphic. Show that also their one-point compactifications are homeomorphic.

Assume $X$ and $Y$ are two homeomorphic locally compact Hausdorff-spaces. Show that also their one-point compactifications are homeomorphic. Give an example of two non homeomorphic locally compact Hausdorff-spaces but which one-point…
JKnecht
  • 6,543
5
votes
1 answer

Example of topological spaces $A \subset X$ with $X \setminus A$ not homeomorphic to $(X/A) \setminus (A/A)$

This is a homework question. I am asked to show that if $A \subset X$ is closed, then $X \setminus A$ is homeomorphic to $(X/A) \setminus (A/A)$. I have done this, and I now have to show by example that this is false if we do not require that $A$ is…
G Pace
  • 315
5
votes
3 answers

Sequences in Non-Hausdorff Spaces

I was told that in any space that is not Hausdorff there are at least two points such that any sequence converges to one iff it converges to the other. I don't know how to prove this. Could I have any help?
Parakee
  • 3,304
5
votes
3 answers

Homeomorphism definition

I was told by my professor that homeomorphisms are continuous maps with continuous inverse, but do those conditions also imply that the map is bijective?
user317935
  • 51
5
votes
1 answer

A question about complete metric spaces.

Is there a theorem which states: "Every infinite metric space that is complete, connected and locally connected, is arc-wise connected"?
Garabed Gulbenkian
  • 1,211
5
votes
1 answer

What are the two disjoint closed sets that cannot be separated by two disjoint open neighborhoods in the Ellentuck topology?

Denote by $X := [\mathbb{N}]^\infty$ the set of infinite subsets of $\mathbb{N}$. Recall that the Ellentuck topology is a topology on $X$ generated by sets of the form $\{A\text{ infinite} \mid s\text{ is an initial segment of }A\text{ and…
Zilin J.
  • 4,270
5
votes
1 answer

Prob. 9 (c), Sec. 18 in Munkres' TOPOLOGY, 2nd ed: Continuity of a map deduced from its restrictions to closed subsets of the domain

Here's Prob. 9, Sec. 18 in Topology by James R. Munkres, 2nd edition: Let $\{ A_\alpha \}$ be a collection of subsets of $X$; let $X = \bigcup_\alpha A_\alpha$. Let $f \colon X \to Y$: suppose that $f | A_\alpha$ is continuous for each $\alpha$.…
Saaqib Mahmood
  • 26,762
5
votes
1 answer

open equivalence relation and closed graph of it

I need to prove that if $\sim$ is an open equivalence relation on a topological space S and $R = \{(x,y)\in S\times S : x\sim y\}$ is a close subset of $S\times S$ then $\Delta = \{(x,x)\in S\times S\}$ is a close subset of $S\times S$. I tried to…
Mathitis
  • 659
5
votes
2 answers

Topology - The arbitrary union axiom

So, the common answer to why we need the concept of topology is that we need it to talk about things like limits of infinite sequences and continuity. But, when we define the axioms of topology, we have an axiom which says that an…
Thiagarajan
  • 657
5
votes
1 answer

Prob. 21, Sec. 17, in Munkres' TOPOLOGY, 2nd ed: Closure and complementation can result in at most 14 different sets

Here's Prob. 21 in Sec. 17 of the book Topology by James R. Munkres, 2nd edition: Consider the collection of all subsets $A$ of the topological space $X$. The operations of closure $A \to \overline{A}$ and complementation $A \to X-A$ are functions…
Saaqib Mahmood
  • 26,762
5
votes
4 answers

$f$ closed iff $y\in N$ and open $V\supset f^{-1}\left(\{y\}\right)$ exists $U$ open such that $V\supset f^{-1}(U)\supset f^{-1}(\left\{y\right\})$

Prove that $f\colon M\to N$ (topological spaces) is closed if and only if for all $y\in N$ and all open sets $V\supset f^{-1}\left(\{y\}\right)$ in $M$ there exists an open set $U$ in $N$ containing $y$ such that $V\supset f^{-1}(U)\supset…
Gaston Burrull
  • 5,449
  • 5
  • 33
  • 78
1 2 3
…
99 100