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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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Does there exist a continuous onto function from $S^2$ to $S^1$?

Here $S^2$ is 2 dimensional unit sphere in $\mathbb{R^3}$ and $S^1$ is unit circle in $\mathbb{R^2}$. Does there exist a continuous onto function from $S^2$ to $S^1$? Till now, I am able to construct a continuous map from $S^2$ to closed unit disk…
Sai Teja
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If two topological spaces have the same topological properties, are they homeomorphic?

Topological properties are investigated because we can show that two spaces are not homeomorphic by finding one property that holds in one space but not the other. But what if no topological property can distinguish two topological spaces? So I…
edm
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Pacman on a Mobius Strip

Pacman lives on the 2D cylinder, $S^1 \times I$, where $I \subset \mathbb{R}^1$ is an interval of the real line. So that we can play the game on a flat surface, we portray Pacman's world on a fundamental domain, a rectangle, and identify two…
Johnver
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Finding the closure of some subsets of the ordered square

I need to find the closure of these sets on the ordered square: $$A = \left\{\left\langle\frac1n,0\right\rangle \mid n\in \mathbb{Z}_+\right\}$$ $$B = \left\{\left\langle1-\frac1n,\frac12\right\rangle \mid n\in \mathbb{Z}_+\right\}$$ $$C =…
Guerlando OCs
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Is saying a set is nowhere dense the same as saying a set has no interior?

I have two statements "A set X is nowhere dense" "A set X has no interior" Are these both equivalent statement?
Al jabra
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Proof for "Given any basis of a topological space, you can always find a subset of that basis which itself is a basis, and of minimum possible size."

The titular statement is used in the explanation of this answer from several years back. I ran across it while puzzling my way through this text, which I bought while I was still in high school and had no idea that I would probably wish I had spent…
Opal E
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Munkres Order Topology example question (Section 14 Example 4)

in section 14, Munkres introduces the order topology, and gives this example: The set $X$ = {1,2} $\times \mathbb{Z}_{+}$ in the dictionary order is an example of an ordered set with a smallest element. Denoting 1 $\times$ $n$ by $a_{n}$ and 2…
Vien Nguyen
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Do non-second-countable spaces have "small" non-second-countable subspaces?

If $X$ is any space which is not second-countable, can one find a subspace $Y \subseteq X$ with $|Y| \leq \aleph_1$ which is also not second-countable? (Recall that a topological space $X$ is second-countable if it has a countable base.) Note that…
Taumatawhakatangihangakoauauot
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Homotopic maps have homotopy equivalent mapping cones

Let $f,g:X\to Y$ be maps of spaces such that $f\simeq g$. Is it true that the mapping cones $\operatorname{cone}(f)$ and $\operatorname{cone}(g)$ are homotopy equivalent? Can we write down an explicit homotopy equivalence? It ought not to be too…
John Steinbeck
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Identifying a map by looking at the pair of topologies that makes it continuous.

Let $\omega_X$ be the set of all topologies on $X$. Given $f:X\rightarrow X$, define $R_f \subset \omega_X \times \omega_X $ as those pairs of topologies on $X$ which make $f$ continuous. For example $\left(\text{Discrete Topology},-\right)$ or…
Hooman
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Find all topology on finite set in especial case

Let $X$ be finite set, Find all topology on finite set with this condition: For every subset $A \subset X$, either $A$ is open or $A$ is closed. I find one of them: $$T=\{ \emptyset, X, \{1\},…
amir bahadory
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Are right continuous functions continuous with the lower limit topology?

Let $f: \Bbb R \to \Bbb R$ be right continuous. Is $f$ continuous as a function from $\Bbb R$ with the lower limit topology to $\Bbb R$ with the standard topology? It clearly seems like it will be, but I'm not sure how to show it.
YAK
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Disjoint union topology vs Product topology

I am trying to understand the differens between the two more precisely. I am aware that the product topology is the product in category while the disjoint union topology is the coproduct. Unfortunately I feel that doesn't quite explain it. If we let…
Zelos Malum
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Real valued function, $M$, $s_n \rightarrow s; |f(s_n)| \le M$ for $s$ in open $U$. (Problem from Gamelin and Greene)

Gamelin and Green, Introduction To Topology Ch. 1 section 3 problem 10. Let $f$ be a real-valued function on $\mathbb{R}$, the real numbers. Show that there exist $M \gt 0$ and a nonempty open subset $U$ of $\mathbb{R} $ such that for any $s \in…
fleablood
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Why is open connected minus countable set is connected?

Reference: Path connectedness of the complement of countable set Let $G$ be an open connected subset of $\mathbb{C}$. And let $E$ be a countable subset of $G$. How do I prove that $G\setminus E$ is path-connected? If $E$ is discrete and has no limit…
Rubertos
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