# 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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### $G_{\delta}$ set

I have a homework as follows: Let $\mathcal{U}$ be an open cover of $X$.If $\mathcal{V}$ is finite subset of $\mathcal{U}$ but $\mathcal{V}$ doesn't cover $X$, prove that $X-\bigcup\mathcal{V}$ is a ...
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### Is the set of solution of $x_0^2+x_1^2-x_2^2=0$ homeomorphic to the set of solution of $x_0^2-x_1^2=0$ in $\mathbb{P}^2(\mathbb{R})$?

Is the set of solution to $x_0^2+x_1^2-x_2^2=0$ homeomorphic to the set of solution to $x_0^2-x_1^2=0$ in $\mathbb{P}^2 (\mathbb{R})$? I think one can show that the first set is homeomorphic to $S^1$,...
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### Properties of a modified Zariski topology

For any set $I \subseteq \mathbb{C}[X_1, \dots, X_n, Y_1, \dots, Y_n]$ of polynomials let us define $$V'(I) := \{ x \in \mathbb{C}^n : f(x,\overline{x}) = 0 \text{ for all } f \in I \}.$$ In analogy ...
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### Limit topology of a sequence of topological vector spaces

Under which circumstances is the limit topology of an increasing sequence $E_0\subseteq E_1\subseteq E_2\subseteq\cdots$ of topological vector spaces, where the inclusion maps are linear and ...
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### $A$ and $A+y$ are homeomorphic where $A$ is open set

Actually I need to understand $A+B$ is open whenever $A,B$ open set in $\mathbb{R}$ First I want to prove $A$ and translation of $A$ by $y,y\in B$ are homeomorphic $f:A\to A+y, f(x)=x+y$ may be the ...
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### basis-free definition of linear function space topology

Let $V$ be a finite-dimensional space over $R$, and $M(V)$ the space of linear operators on $V$. I can choose an ordered basis in $V$ and identify it with $R^n$, and identify $M(V)$ with $R^{n^2}$, ...
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### Topology with equivalence of convergence of nets and almost everywhere convergence

I want to show that there is no topology for the set of Lebesgue measurable functions such that the net $<f_n> \to f$ iff $f_n \to f$ almost everywhere. Assume that there exists such a topology....
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### Union over disjoint union

How does the normal union behave over the disjoint union? For instance, if i have some indexed collection of disjoint unions between two sets, what is the union over the whole collection?
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### Find $A^\circ , \, cl (A),\, A', \partial A$

Consider the set $$Α=\left\{ \left(\dfrac 1 n, \dfrac 1m \right):\, n,m \in \mathbb N\right\}.$$ We want to find the sets $A^\circ=int\, A, \,cl(A) ,\, A' (\text {= derived set}) , \, \partial A$. ...
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### Complement of contractible subset of a sphere

Let $A$ be a nice closed subset of the sphere $S^n$; for example, we could ask $A\to S^n$ to be a cofibration. Assume that $A$ is contractible. Is then $S^n - A$ also contractible? It ...
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### Using Baire Category Theorem to prove $\mathbb{R}^2\not\cong\mathbb{R}^3$.

How can we prove $\mathbb R ^ 2$ is not homeomorphic to $\mathbb R ^3$ using Baire Category Theorem? Here is a standard proof of this fact using algebraic topology. Note that $\mathbb{R}^{3}-\{x\}$ ...
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### Norms on $\mathbb{R}$ seen as a $\mathbb{Q}$-vector space.

this is not really a question : I had some ideas on topics I don't feel secure with. I expose these hereafter : are there any mistakes in my reasonning ? Also, if anyone knows a good read about this ...
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### A question about the Skorokhod topology

I have a question which may be naive but I can not find the answer in general reference about Skorokhod topology. Let $\{w_n\}_{n\ge 0}$ be a sequence of cadlag functions defined on $[0,1]$ such ...
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### Hausdorffness of quotient space

Let $G$ be a compact topological group, and $X$ be a Hausdorff space. We assume that $G$ acts on $X$. Is the quotient space $X/G$ with the quotient topology a Hausdorff space? It seems that the ...
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### Discrete Analogue of the Poincaré Conjecture and Simple Connectedness

I apologize if this question is badly worded or obvious, but I have no formal topology background. I have put some effort into trying to find something, but nothing turned up, perhaps due to my lack ...
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### Path Homotopy in a Topological Annulus

Let $C_1$ and $C_2$ be simple, closed curves in $\mathbb{R}^2$ such that $C_1$ lies in the region bounded by $C_2$, and the origin $O$ lies in the region bounded by $C_1$. Define an annulus $A$ as the ...
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### How many Y shapes can you fit on the plane?

You can only fit at most countably many disjoint open discs on the plane: for any collection of disjoint open discs, it is possible to pick a single rational coordinate contained in each disc, and ...
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### Introductory Topology True/False check - Topology without tears - Exercises 1.1

I have just started to learn Topology, using specifically the book mentioned in the title. I have placed that information in the title with SEO in mind, if this is not acceptable practice in this ...