# Tagged Questions

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; ...

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### Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
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### Initial topology of the spectrum mapping $\sigma$

Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$. The collection of compact subsets ...
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### Determining whether a path must pass through a given set

I have sets $E^1, \dots, E^n \subset \mathbb{R}^n_{\ge 0}$, none of which contain $0$. I would like to determine whether or not these sets form an "overhang" of $0$. Intuitively speaking, this is an ...
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### Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
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### Fiber Bundle: Hairbrush

I am trying to understand the hairbrush example of a fiber bundle from the Wikipedia article on fiber bundles. If I am understanding this, in the hairbrush example E is the hairbrush, ie. all the ...
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### Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one ...
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### $\sigma$-product space and $\Sigma$-product space

Recently, I'm interested in the $\sigma$-product space and $\Sigma$-product space. Is there a survey on $\sigma$-product space and $\Sigma$-product space, which is simple for a beginner? Thanks for ...
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### Smooth deformation retracts

Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For ...
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### Countable Product of discrete spaces

Let $X$ be a countable discrete topological space. Consider $X^{\mathbb{N}}$ endowed with the product topology. How do you prove that $X^{\mathbb{N}}$ is homeomorphic to the sub-space of all ...
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### Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...
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### uniform distribution on unit ball

If $S$ is a set of an countably infinite number of points uniformly distributed throughout the unit ball in $\mathbb R^n$, is there for every point $p$ in the ball and every real number $e>0$, a ...
I'm going through Spanier and got stuck on the following problem: Show that a space $Y$ is contractible if and only if given a pair $(X,A)$ having the homotopy extension property with respect to $Y$, ...