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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
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Show that something is not a retract

I am trying to prove that ($S^1\times\{1\})\cup(\{1\}\times S^1)$ is not a retract of $S^1\times S^1$. Any help would be appreciated. Thank you!
jmcats
  • 141
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Prove that a space having the fixed point property must be connected.

A space X is said to have the fixed point property if every map $f: X \to X$ has a fixed point, i.e. a point $x_0 \in X$ such that $f(x_0) = x_0$. Prove that a space having the fixed point property must be connected. completely stuck on it.can…
TUMO
  • 51
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Topologically, how does a "super continuum" differ from the reals?

This is a question I've wondered about for a while. Consider the following setup. Suppose you take a non-Archimedean ordered field, a strict field extension of the reals, and then you form its Dedekind completion. We are interested in these…
The_Sympathizer
  • 19,651
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Is the Katětov extension of $\Bbb N$ zero-dimensional?

I take $X=\kappa\Bbb N$ to be $\Bbb N\cup\{p:p \text{ is a free ultrafilter on }\Bbb N\}$. Each singleton in $\Bbb N$ is open and a local base at any free ultrafilter $p$ is given by $\{\{p\}\cup A:A\in p\}$. Is X zero-dimensional? In other…
PatrickR
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Closedness with respect to an open cover

This is something that came up when I was studying something else, but I am wondering whether the following topological fact is true. Let $X$ be a topological space, and $\{U_i\}_{i=1}^n$ a finite open cover of $X$. Let $A \subseteq X$ be a…
JHF
  • 10,996
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Separating points in the plane by a compact set that is not allowed to intersect a given set A

Suppose $A\subset \mathbb{R}^2$ is closed and totally disconnected, and suppose $a,b \in A$. Is it possible to find a compact subset $C$ of the plane, which is disjoint from $A$, and which separates $a$ and $b$? (That is, $a$ and $b$ are in…
alephzero314
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Is convex open set in $\mathbb{R}^n$ is regular?

Convex set $A$ in $\mathbb{R}^n$ iff for every $ x,y\in A,0\le\lambda\le1$, then $\lambda x+(1-\lambda)y\in A$. $U$ is regular open set iff $A$ is open and $int(cl(A)) = A$. Is convex open set in $\mathbb{R}^n$ is regular?
Leitingok
  • 2,790
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Help in understanding product topology

Let $X_i$ be topological space. Let $O_i \subseteq X_i$ denote open set and let $C_i \subseteq X_i$ denote closed set. Let $O$ be open set in $X = \prod_i X_i$ and let $\pi_i : X \to X_i$ be projection map. Then $O = \bigcap_{i=0}^n \pi_i^{-1}O_i$…
blue
  • 2,884
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For each continuous $g:X\to [0,1]$, $g(a_n)\to g(a)$, can we deduce $a_n\to a$?

Assume $(X,\mathcal T)$ is a Tychonoff space. $(a_n)$ is a sequence in $X$. $a\in X$. for each continuous function $g:X\to [0,1]$, $$g(a_n)\to g(a)$$ Is there an elementary proof for $$a_n\to a$$ ?
user59671
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About interior, closure and etc in $\mathbb{R}^2$

Given this set $A$: $$A =\left\{ \, (x,y) \mid x = 1/n, \ |y| \le n, \ n \in \mathbb{N} \, \right\} \subset \mathbb{R}^2; $$ I'd love to find the interior, closure, set of limit points, set of contour points (I doubt this is a correct English term)…
Pranasas
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How to prove that the surface of genus $2$ can be represented as an octagon

I've been trying to show that the surface of genus $2$ can be represented by appropriately identifying the edges of a regular octagon. I think I have managed to work out the way to identify the edges but how can I prove that it is indeed of genus…
user61496
  • 483
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What topological structures have exactly one base?

for homework What topological structures have exactly one base? I think that: $\{\emptyset,X\}$ his basis is $\{\emptyset,X\}$ is this ok? I am not sure, could please provide me another example and why?
LFRC
  • 643
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Doubts on the mathematical content of the Hitler Downfall Meme.

This question was inspired by this "beautiful mind" question, where some one asks about the mathematical contents of some movie. I do not see how this other question is less opinion-based than the one I am posing. This post also did not motivate…
André Caldas
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Can convergence be seen as a form of continuity?

I will abbreviate "it holds that" to "iht" and "such that" to "sth." The following questions are motivated by curiosity. Question 1. Does there exist a topology on $\mathbb{N}$ such that for all topological spaces $Y$ and all sequences $a :…
goblin GONE
  • 67,744
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Is it in general true that a space is not homeomorphic to the punctured version of this space?

For non-arbitary spaces we can discuss for such case, like how many components are there or other properties. But is it true for any space? It seems if we have a homeomorphism $f$ from $S$ to $S' = S - \{p\}$, $f(p) = q$, but since a space is…
MonkeyKing
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