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
7
votes
1 answer

Existence of open set in product topology

Let $X$ be a compact topological space and $Y$ a Hausdorff space. Let $C \subseteq Y$ be closed in $Y$ and $U$ an open set in $X \times Y$ which contains $X \times C$. Prove there exists an open set $V \subseteq Y$ such that $X \times C \subseteq X…
user
7
votes
1 answer

Compact sets in the product of topological spaces.

Let $G_1$ be a non-compact topological space and let $G_2$ be a generic topological space. What are the compact sets in the product $G_1\times G_2$? Surely we can take the sets of the form $K_1\times K_2$ where $K_1$ and $K_2$ are respectively…
Richard
  • 1,012
  • 8
  • 18
7
votes
2 answers

Tweaking the axioms of a Topological Space, what are the consequences?

A topological space is a set $X$ together with a topology $\tau$ (a collection of open subsets) such that. $\emptyset\in \tau$ and $X\in \tau$. The intersection of a finite number of sets in $\tau$ is also in $\tau$. The union of an arbitrary…
user153330
  • 1,726
7
votes
3 answers

Topological Proof that every Interval $I \subset \mathbb{R}$ is connected

First, the definition of a connected set: Definition: A topological space is connected if, and only if, it cannot be divided in two nonempty, open and disjoint subsets, or, similarly, if the empty set and the whole set are the only subsets…
StefanH
  • 18,086
7
votes
1 answer

Accumulation points / Cluster points / Closed sets

In a topological space $X$, call $x\in X$ an accumulation point if $\forall$ open set $U\ni x$, $U \cap A \neq \emptyset$, and $y\in X$ a cluster point if $\forall$ open set $U\ni y$, $U\cap A\setminus \{y\} \neq \emptyset$. (These are the…
hwhm
  • 729
7
votes
2 answers

Compact Hausdorff Spaces with pre-caliber $\aleph_1$ has caliber $\aleph_1$

Let us recall that a topological space has $\aleph_1$ pre-caliber (resp. caliber) if given any family of open sets $\{U_\alpha\}_{\alpha<\omega_1}$ there exists an uncontable set $B\subset\omega_1$ such that the subfamily $\{U_\alpha\}_{\alpha\in…
Ergonvi
  • 589
7
votes
1 answer

Closure of the interior of another closure

Let $X$ be a topological space and let $A \subset X$. Is it true that $\overline{\rm{Int}(\overline{A})}=\overline {A}$? This question arose when I try to show$\overline{X-\overline{\rm{Int}(\overline{A})}}=\overline{X-\overline{A}}$
user31899
  • 3,917
  • 25
  • 53
7
votes
1 answer

Product space that is compact, but isn't sequentially compact

I need to find an example of infinite product space that is compact, but it's not sequentially compact. So this is my example: $$ \prod_{i\in [0,1[}\{0,\ldots,9 \}. $$ It sure is compact, but is it sequentially compact? Let's define a sequence…
Zzz
  • 691
7
votes
1 answer

Basis elements of the uniform topology on $\mathbb R^{\omega}$

I have been trying to understand the basis elements of the uniform topology on $\mathbb{R}^{\omega}$. For some time, I thought they would be: $B_\bar{p} (x,\epsilon) = \prod (x_i - \epsilon, x_i + \epsilon)$ if $\epsilon < 1$ However, after reading…
Avatrin
  • 1,527
7
votes
1 answer

The homeomorphism $D^n/S^{n-1}\cong S^n$

I want to show that $$D^n/S^{n-1}\cong S^n$$ Let $p$ be the north pole of $S^n$ and denote $(D^n)^o$ the interior of the disc and let $s:\mathbb{R}^n\rightarrow S^n$ be the stereographic projection. Let the map $f:D^n\rightarrow…
palio
  • 11,064
7
votes
4 answers

Can a set be neither open nor closed?

Can a set be neither open nor closed? An example would do. I cant think of any. Thanks in advance!
Jennie Durham
  • 535
7
votes
2 answers

When are polynomial maps open or closed?

1.$f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=x^4+x^3$ is a closed map(or an open map)? 2.$f:\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=x^5+x^4$ is a closed map? 3.$f:\mathbb{R}^2 \rightarrow \mathbb{R}$, $f(x,y)=x^2+y^2$ is a closed map? Generally,…
David Chan
  • 1,960
7
votes
2 answers

sequence of decreasing compact sets

In Royden 3rd P192, Assertion 1: Let $K_n$ be a decreasing sequence compact sets, that is, $K_{n+1} \subset K_n$. Let $O$ be an open set with $\bigcap_1^\infty K_n \subset O$. Then $K_n \subset O$ for some $n$. Assertion 2: From this, we can easily…
XxXxX
  • 357
  • 2
  • 9
7
votes
1 answer

What is the "Tychonoffication"?

In this link: https://mathoverflow.net/questions/23940/why-free-topological-groups-on-tychonoff-spaces I read the following: Let $X$ be a topological space. The Tychonoffication $Y$ of $X$ is the quotient of $X$ by the relation $x\sim y$ iff…
Tanius
  • 1,195
7
votes
1 answer

Does the 'closure of the interior' equal the 'interior of the closure'?

My answer is no because, $\mathbb{Q}^o = \emptyset$ and so $\overline{(\mathbb{Q}^o)} = \emptyset$ but $\overline{\mathbb{Q}} = \mathbb{R}$ and so $\big(\overline{\mathbb{Q}}\,\big)^o = \mathbb{R}$. Is my example correct?
user26069
1 2 3
…
99 100