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.
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How to prove the continuity of the path lifting function of covering spaces.
Let $p : E \to B$ be a covering map, the path lifting function $\varphi : E \times_p B^I \to E^I$, where $E \times_p B^I := \{(e, \gamma) \in E \times B^I : \gamma(0) = p(e)\}$ is a pullback of $E$ ...
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A surjective continuous open map has a continuous right inverse
My question arises from this post and similar questions. It's clear that for an open, onto and continuous function $f:X\rightarrow Y$, not every right inverse is continuous, but my question is if ...
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Is there a continuous bijective mapping of $R$ into a compactum
I wonder if there are generally no bijective, continuous mappings of the form $$f: \mathbb{R} \rightarrow K$$ if $K$ is a topological compact space.$$$$ My considerations I realize that the inverse ...
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Looking for an example of a real (hyper)cubic surface with special restriction
Need help looking for an example of a cubic surface over $\mathbf{R}$: it has two connected components, one of the components is convex. Here's my thoughts so far:
The model in my head is the ...
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When will "Retract $\iff$ Deformation Retract" hold true?
Problem. Given topological space $X$ and subspace $A\subset X,$ under what conditions will we be able to make the following claim: $A$ is a retract of $X$ if and only if $A$ is a deformation retract ...
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When must a space generated by compacts also be generated by Hausdorff compacts?
I'm interested in comparing $k_1$-spaces,
spaces whose topologies are witnessed by
their compact subspaces, and $k_3$-spaces,
spaces whose topologies are witnessed by their
compact and Hausdorff ...
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Book Recommendation: Topology & Urelements [closed]
I am looking for a book about Topology with urelements. I am wondering whether the matter has ever been researched in the first place, especially since I could not find any trace on the internet: not ...
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How can this open map characterization be explained or interpreted?
I hope you're having a good day.
I'm an undergrad mathematics student, I took a general topology course a few months ago, and I'm now reviewing topological spaces to prepare for functional analysis.
I ...
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Is this relation making a space Hausdorff?
Given a topological space $X$ there are ways of introducing relations $\rho$ on it so that $X/\rho$ becomes a Hausdorff space. Here I will describe two relations which I will call inner and outer (...
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Compact in the relative topology [duplicate]
Quick simple question, but I want to make sure I am correct: Is the set $(0,1]$ compact in $(0, \infty)$ endowed with the relative topology of standard topology in $\mathbb{R}$? Note that I am aware ...
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Homotopy equivalence of the tetrahedron minus its interior
Let's consider a tetrahedron $T$. Let's remove the interior and the interior of three of its faces. What space is it homotopy equivalent to?
My attempt.
Suppose that the base of the tetrahedron is the ...
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Closed sets definition in complex analysis [closed]
Here's the translation:
Are limit points and boundary points the same? If not, why are there two versions of the definition for a closed set? Which one is correct?
A set $A \subset \mathbb{C}$ is ...
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Riemannian Manifolds (Lee) Exercise 6-28 (d): Isometries converging pointwise converge topologically
Suppose $M$ is a connected, complete Riemannian manifold, $\mathrm{Iso}(M)$ is a smooth Lie group composed of all isometries of $M$, and $\phi_n$ is a sequence of isometries converging to an isometry $...
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For compact Hausdorff spaces, is countable pseudocharacter equivalent to first countable? [duplicate]
Let $X$ be a compact $T_2$ space. Is $X$ first countable if, and only if, $X$ has countable pseudocharacter?
Note: I have already proven that every $T_1$ first countable space has countable ...
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Manifolds and covering maps [duplicate]
I am studying topological manifolds; I know that if $M$ is an $m$-topological manifold and $p:M\rightarrow N$ is a covering map, then $N$ is an $m$-manifold. I know how to prove the existence of local ...
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