# 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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### Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
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### Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd). Questions: Suppose that we ...
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### Topological proof that this set is a topological manifold

let $S \subseteq \mathbb R^3 \times \mathbb R^3$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show that this is a topological manifold without ...
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### Find the closure for several sets

(a) $\mathbb{Q}$ (b) {$(x,y)\in\mathbb{R}^2:xy<1$} (c) {$(x,\sin($${1}\over{x}$$)):x>0$} (d) {$(x,y)\in\mathbb{Q}^2:x^2+y^2<1$} First Closure $\overline{A}$, it is a set contains all ...
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### Topological inequivalent manifolds obtaining by removing a surface from a manifold

Are there any general techniques for classifying the inequivalent topologies that can be obtained by removing a 2-surface S from a 4-manifold M? I am particularly interest in the case where both M and ...
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### Pointfree generalization of uniform spaces?

Topological spaces generalize as frames and locales. But are there a pointfree generalization of uniform spaces?
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### Euler characteristic of structure sheaf of symmetric product

I recently asked about calculating the Euler characteristic of the symmetric square of a space. There we determined that for a sufficiently well-behaved space $X$ there is a formula \chi(X \times ...
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### Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?
Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...