# Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

2,953 questions
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### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with ...
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### How many polynomial Morse functions on the sphere?

Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function. If $f$ is a Morse function of degree $1$, you ...
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### Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$. As a corollary of something else I was playing around with, I recently ...
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### Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?

This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable ...
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### Topology on the set of analytic functions

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on ...
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $... 3answers 661 views ### What sets of self-maps are the continuous self-maps under some topology? An open question on MSE, https://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a ... 1answer 558 views ### Which ordered fields are homeomorphic to their power? It is well known that$\mathbb{R}^2\ncong \mathbb{R}$. It is also known that$\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ... 8answers 3k views ### Connections between ultrafilters in topology and logic I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is ... 4answers 2k views ### The role of ANR in modern topology Absolute neighborhood retracts (ANRs) are topological spaces$X$which, whenever$i\colon X\to Y$is an embedding into a normal topological space$Y$, there exists a neighborhood$U$of$i(X)$in$Y$... 2answers 1k views ### Is every closed set of Q² the intersection of some connected closed set of R² with Q² Let$F\subset\mathbb{Q}^2$a closed set. Does there exists some closed and connected set$G\subset\mathbb{R}^2$such that$F=G\cap\mathbb{Q}^2$? For example if$F=\{a,b\}$, you can take$G$the ... 2answers 896 views ### Which are the rigid suborders of the real line? Which are the rigid suborders of the real line? If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ... 3answers 866 views ### Possible categorical reformulation for the usual definition of compactness Let$X$be a compact topological space,$f_i:Y_i\to X$a family of continuous maps such that the topology on$X$is final for it (i.e.,$U\subset X$is open iff$f_i^{-1}(U)$is open for each$i$, for ... 4answers 898 views ### Spaces with no topological monoid structure which are homotopy equivalent to topological monoids In motivating$A_\infty$-spaces to my students I'm going to insist on the homotopy invariance of the notion, saying that "being$A_\infty$is the homotopy invariant version of being a topological ... 6answers 2k views ### Is there a topological description of combinatorial Euler characteristic? There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ... 4answers 1k views ### When does a Galois connection induce a topology? Let$(X,\leq)$and$(Y,\leq)$by partially ordered sets. Recall that a(n antitone) Galois connection between$X$and$Y$is a pair of order-reversing maps$\Phi: X \rightarrow Y, \ \Psi: Y \...
It seems like when we assume "niceness" in homotopy theory we assume that $X$ has the homotopy type of a CW complex, and in fiber bundle theory we assume that $X$ is paracompact. How do these two ...