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Questions tagged [gn.general-topology]

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

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A consecutive resolution of continum algebras to a simple continum algebra

Motivated by classical Gelfand Naimark duality, the correspondence between the category of commutative $C^{*}$ algebras and the category of locally compact Hausdorff spaces, we ...
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104 views

Continuous functions “sharing” a point

Let $(X,\tau)$ be a topological space. By $\text{End}(X)$ we denote the collection of all continuous functions $f:X\to X$. We say $f,g\in \text{End}(X)$ share a point if there is $x\in X$ such that $f(...
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1answer
162 views

Continuous real function on germs

Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$...
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2answers
369 views

Homotopy problem for infinite dimensional topological space

Let $X$ be an infinite dimensional topological space such that : $ \forall n \in \mathbb{N}$, $ \exists X_{n} \subset X$, $n$-dimensional subspaces verifying : $\forall r<n$, the homotopy ...
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1answer
320 views

Is there a continuous surjection $\omega^\omega\to \mathbb{R}$? [closed]

Let $\omega$ be endowed with the discrete topology, and let $\mathbb{R}$ carry the Euclidean topology. Is there a continuous surjective map $f:\omega^\omega\to \mathbb{R}$? (I suppose this would ...
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1answer
172 views

Does order-preserving equal continuous? [closed]

Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?
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1answer
132 views

Does every compact countable space contain a non-trivial convergent sequence?

Problem. Does every compact countable space contain a non-trivial convergent sequence? This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed ...
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192 views

A noncommutative analogy of the tube lemma

Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to \mathbb{C}\...
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1answer
110 views

Can we build a continuous function from “fibers”/preimages defined over a topological base?

I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
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1answer
63 views

$T_2$-spaces such that the lattices of open sets can be embedded into each other

Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$. Does this imply that $(X,\tau)\cong (Y,\sigma)$?
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space

Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
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1answer
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Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$

Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by $$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$ where $\downarrow x = \{y\in Q: y\leq ...
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1answer
137 views

Bounded metric spaces with non-surjective self-isometry

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
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1answer
377 views

A topological groupoid structure on a pair $(X,A)$

Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$. Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding ...