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
4 answers

Set of infinite subsets. Is it a topology?

Here is the question: Let $X = \mathbb{R}$ and let $\Omega$ consist of the empty set and all infinite subsets of $\mathbb{R}$. Is $\Omega$ a topological structure? My attempt : I think the answer is No; it is not a topology. Because we have $$…
Rusty
  • 2,868
6
votes
1 answer

$\coprod_{i \in I} (Y \times X_{i})$ and $Y \times \coprod_{i \in I}X_{i}$ are homeomorphic

To prove that $\coprod_{i \in I} (Y \times X_{i})$ and $Y \times \coprod_{i \in I}X_{i}$ are homeomorphic, I constructed $h: \coprod_{i \in I} (Y \times X_{i}) \rightarrow Y \times \coprod_{i \in I}X_{i}: ((y,x),i) \longmapsto (y,(x,i)).$ Now it is…
KarenVO
  • 315
6
votes
2 answers

Density of a dense subspace of a Hausdorff space

If X is a Hausdorff space and Y is a dense subspace of X, can the density of Y exceed the density of X? The density of a space X is the least infinite cardinal C such that X has a dense set of cardinal C or less.
DanielWainfleet
  • 57,985
6
votes
2 answers

$A \subseteq \mathbb{R}^n$ closed and connected. Prove $\{x \in \mathbb{R}^n \mid d(x, A) \le \varepsilon\}$ is path-connected

I've encountered the following question: Say $A \subseteq \mathbb{R}^n$ is a closed and connected set. Prove $\{x \in \mathbb{R}^n \mid d(x, A) \le \varepsilon\}$ is path-connected. I'm not really sure how to approach this question. It appears under…
amirbd89
  • 1,052
6
votes
2 answers

Weakening paracompactness condition

Let $X$ be a topological space such that every open cover has a finite refinement. Then is $X$ compact, or is there a counterexample? Let $X$ be a topological space such that every open cover has a locally finite subcover. Then is $X$ compact, or…
Marketa
6
votes
1 answer

Extend by Continuity

This is a very short question, I hope it is not too broad, if so I shall try and make it more specific. I would like to start as it stands below, though, because it really points down the essence of the question: What do people mean when they write…
harlekin
  • 8,740
6
votes
1 answer

example of a open function such that the restriction is not open

Give an example of a function $f:X \to Y$ and a subset $A \subset X$ such that f is open but $f_A$, the restriction of $f$ to $A$ is not open. Can someone help me please? Thanks
Ryoma
  • 117
6
votes
3 answers

Neighborhoods vs Open Neighborhoods?

Since when I started studying general topology there is something concearning neighborhoods which baffles me. Given a topological space $(X, \mathscr{T})$ and $p\in X$ we say $U\subseteq X$ is a neighborhood of $x$ if there is $W\in\mathscr{T}$ such…
PtF
  • 9,655
6
votes
1 answer

Is the complex plane homeomorphic to $\mathbb{R}^2$?

Is the set of complex numbers homeomorphic to $\mathbb{R}^2$? They are isomorphic. Are they homeomorphic?
Junior Soares
  • 537
6
votes
3 answers

Is the boundary of a boundary a subset of the boundary?

The definition of a boundary of a set $S$ in a topological space $X$ is $\text{comp}\{\text{Int}(S) \cup \text{Ext}(S)\}$ (complement of the interior union exterior). The definition for interior is the set of all interior points. The definition for…
mr eyeglasses
  • 5,539
6
votes
1 answer

Topology on the extended real line

In our real analysis class we are working through 'Real and Complex Analysis' by Rudin and covered topological spaces (but not bases, subbases and other ways of generating topologies, so I can't use these in the exercise). Let $\tau =$ the…
Olorun
  • 1,552
6
votes
1 answer

Connected subspaces question

Suppose that C, D are connected subsets of a topological space T such that $\bar{C} \cap \bar{D} \neq \emptyset$. Is it true that $C \cup D$ is necessarily connected? I think I have a counter example for this: Take the intervals $C = (0,1)$ and $D =…
user26069
6
votes
1 answer

Question from Munkres Topology Regarding the Product Topology (Section 16, Exercise 5)

Let $X$ and $X'$ denote a single set in the topologies $\mathscr T$ and $\mathscr T'$ respectively; let $Y$ and $Y'$ denote a single set in the topologies $\mathscr U$ and $\mathscr U'$ respectively. Assume these sets are non-empty. a) Show that if…
Tim Raczkowski
  • 20,046
6
votes
3 answers

How many of these are topologies?

Let $X$ be a set with $3$ elements. The set of subsets of the power set of $X$ is $2^{2^3}$ elements. How many of these are topologies? Is there a trick to this problem, or is it just a "plug and chug" casework thing?
user208492
6
votes
3 answers

Urysohn's Lemma: Proof

Given a normal space $\Omega$. Then closed sets can be separated continuously: $$h\in\mathcal{C}(\Omega,\mathbb{R}):\quad h(A)\equiv0,\,h(B)\equiv1\quad(A,B\in\mathcal{T}^\complement)$$ Especially, it can be chosen as a bump: $0\leq…
C-star-W-star
  • 16,275
1 2 3
…
99 100