# 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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### Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
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### Product varieties with the constructible topology

Let $k$ be an algebraically closed field and let $X\subseteq k^n$, $Y\subseteq k^m$ be two affine algebraic varieties. It is not difficult to find examples where the Zariski topology on the product ...
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### From Jordan's Curve Theorem to Jordan-Schoenfliess theorem

I am trying to learn and understand proofs of classical theorems and successfully mastered a proof of JCT. (It was the well-known proof that uses Tietze Extension and Brouwer's fixed point theorem). ...
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### Proof that a set $X \subset M$ is a Manifold

Let M be a manifold without boundary and let , $g:M\to \mathbb R$ have $0$ as a regular value. Than the set $X \subset M$ with $g(x) \ge 0$ is a smooth manifold with boundary equal to $g^{-1}(0)$. I ...
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### Orientability as a topological property

Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such ...
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### Construction of Lakes of Wada

At each step of the construction of Lakes of Wada we extend a lake (an open set in the open unit square) so that no point of the land (the complement of all the lakes) is farther than a given small ...
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### Quotient of complete linearly topologized ring

The quotient of a complete metrizable group by a closed normal subgroup is always complete, but there are examples to show this need not be true for non-metrizable groups. Here complete means every ...
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### Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
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### Compact Hausdorff implies product of quotient map is a quotient map?

Let $X$ be compact Hausdorff and let $q : X \to Y$ be a quotient map. Is it true that $f : X \times X \to Y \times Y$ with $(x_1, x_2) \mapsto (q(x_1), q(x_2))$ is a quotient map?
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### Isotopy between two open disks on a surface

So I have a (compact) surface $\Sigma$ and two open disks on the surface call them $A$ and $A'$ such that the intersection contains a simple curve $P$. What I want to do is construct an isotopy ...
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### Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
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### Counter-example about paracompactness

I am trying to find a counter-example related to the definition of paracompactness, but it seems that it is not very easy. Here is the problem. Give an example to show that if $X$ is paracompact, ...
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### Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
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### Hartshorne II Prop 2.6

Prop 2.6 constructed a continuous map $X$ to $t(X)$, I cannot verify that it is a homeomorphism. I try to show any open set $U$ is mapped to $t(X)\setminus t(X\setminus U)$. To show it is surjective, ...
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### Looking for articles on postcritically finite rational maps in Russian or French

I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French.
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### Is this case possible (hedgehog metric, colinearity)

My topology class was asked to prove that the hedgehog metric was indeed a metric (the details are irrelevant for my question). This does not concern the proof itself, but rather the structure of the ...
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### Not 1-dimensional homological equivalent of the circle

The questions origins from this problem and my incorrect answer to it. I'm trying to correct it, but it turned out that the topological space - that I need to do it straightforward - has very specific ...
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### What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...
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### Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection). A topological space $X$ is linearly ...
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### Is this proof correct: domain of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected.

The domain $X$ of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected. Proof: If $X = F \uplus G$ for two nonempty closed sets $F,G$ ...
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### Union of Sets in Locally Compact Hausdorff Space

Is it possible for an open set in a locally compact Hausdorff space to not be the union of an increasing sequence of compact sets? If so, given a regular Borel measure on such a space, how is it that ...
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### Prove equivalent metric spaces

Let $X_1=[1,2]$ and $X_2=[0,1]$. Let $d_1$ denote Euclidean and let $d_2(x,y)=2|x-y|$ in $X_2$. Show that $(X_1,d_1)$ and $(X_2,d_2)$ are equivalent metric spaces. How do I do that?
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### Can you construct a coutable local base in the space of continuous functions?

Let $(C,\tau)$ be the topological vector space of all complex continuous functions on $[0,1]$ with seminorms $p_x(f)=|f(x)|$, $x\in [0,1]$. We have known $(C,\tau)$ is not metrizable,but how could I ...
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### Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?
Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. $X$...
A space $X$ is constructed from two disjoint copies of $RP^3$ and a copy of the unit interval $I$ by gluing one end of $I$ to a point of one copy of $RP^3$, and gluing the other end of $I$ to the ...