# Questions tagged [gn.general-topology]

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

414 questions
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### A function composed with itself produces the identity

Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...
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### Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow. In non-Hausdorff topology it is standard to ...
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### Spectra of $C^*$ algebras

Gelfand-Naimark structure theorem for $C^*$ algebras gives a canonical isometric * isomorphism between any commutative unital $C^*$ algebra $A$ and the algebra of continuous complex-valued functions ...
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### Generalizations of the Tietze extension theorem (and Lusin's theorem)

I am reasking a year-old math.stackexchange.com question asked by someone else. (For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.) The Tietze ...
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### Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?

Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
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### A generalization of residual finiteness to topological groups

Consider the following generalization of residual finiteness to topological groups. A locally compact Hausdorff group $G$ is called residually compact if for every compact $K \subseteq G$ there is a ...
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### Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...
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### Trivial fiber bundle

Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial? A non connected counter example has been provided, so I'll ask for E,B ...
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### Are topological fiber bundles on the same base with homeomorphic fibers isomorphic?

First I apologize because this is not a research question, but I can't get any answer on MathStackExchange... Let $\pi \colon E \to B$ and $\pi' \colon E' \to B$ two topological fiber bundles on the ...
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### When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)

The title has it all. I'm looking for a reference to the following: Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
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### Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.

Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$ Let us define the following: Condition 1: $X$ can be covered by ...
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### Category of Uniform spaces

I suspect that the category of uniform spaces and uniformly continuous maps and the full subcategory of complete uniform spaces are both bicomplete and cartesian closed. Can anyone comfirm or deny, ...
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### Topological fraction rings and fields

Linked to this question and as a sequel to my answer of it. Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$. Let  s_{frac}\ :\ R\times S\to S^{-...
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### Does geometric realization commute with passing to the compactly generated topology?

My question is in the title, but here is a more detailed formulation: Let Top be the category of all topological spaces and continuous maps, and let CGTop be the subcategory of compactly generated ...
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### On locally convex (and compactly generated) topological vector spaces

Part 1: How big is the category $TVS_{loc.conv.}$ of locally convex topological vector spaces (and continuous maps)? In other words (and less cheekily), is there a free locally convex TVS having any ...
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### On continuous perturbations of functions of the first Baire class on the Cantor set

Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
### another question about connected open sets in $R^2$
Before posting this question,I just asked a similar question:a question about connected open sets in $R^2$. I got several nice answers.Now I want to ask: Let $U$ be a nonempty connected open set in \$...