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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2
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67 views

$G_{\delta}$ 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 ...
2
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25 views

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 $S^1$,...
2
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35 views

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 ...
2
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142 views

the continuity of argmin

how can I show the minimum of a converging sequence of continuous functions is converging to the minimum of the limit of that sequence.i.e. $$\lim_i \text{argmin}_{x} f_{i}(x)=\text{argmin}_{x} \lim_i ...
2
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57 views

Proving $S^{4}/G$ is simply connected where $G$ is not a free group action

Consider the sphere $S^{4}$ as a subset of $\mathbb{R}^{5}$ and consider the action of the group $G$ of homeomorphisms generated by $(x_1, x_2, x_3, x_4, x_5) \rightarrow (-x_2, x_1, -x_4, x_3, x_5)$ ...
2
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76 views

Is locally compact , connected metric space $\sigma$-compact?

Is locally compact , connected metric space $\sigma$-compact? There is a proof in Spivak's book : Differential Geometry VOL 1, page 5. But unfortunately I can not follow it. Does there exist some ...
2
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98 views

Categorical description of Alexander horned sphere?

Looking at the construction of the Alexander horned sphere: Is it possible to describe as some kind of direct limit of spaces? How can one formally see this?
2
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100 views

ANR is locally contractible

Recall that a space $X$ is contractible if there exists a homotopy $h:X\times [0,1]\to X$ such that $h$ is equal to the identity map on $X\times\{0\}$ and $h$ is constant on $X\times\{1\}$. Please ...
2
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124 views

Two disjoint connected and bounded open sets in the plane that shares the same boundary

In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary. My attempt: Since both my sets need to be bounded, my ...
2
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68 views

Dual of path in a space.

Is there a notion dual to the notion of a path in a topological space? Given that a path in a space $X$ is a continuous function from the interval $[0, 1]$ to X, what would the dual of this notion be, ...
2
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67 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I noticed,...
2
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49 views

Van Kampen Theorem for a Certain Square

Take a square with all the edges identifies. Choose a point $x$ on the boundary of this square. Take a small neighborhood $B_\epsilon(x)$ of this point. I want to compute $\pi_1(B_\epsilon(x) \...
2
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96 views

Compact $G_\delta$ subsets of locally compact Hausdorff spaces

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a ...
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199 views

Showing that an evaluation map is continuous

This is a problem from Munkres' Topology 43.8 If $X$ and $Y$ are spaces, define e : $X \times \mathscr {C}(X,Y) \to Y$ by the equation e($x,f$) $= f(x)$; the map e is called the evaluation map. ...
2
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75 views

Why $(\cos t, \sin t)$ is not a homeomorphism

Consider the map $f: [0,2 \pi ) \to S^1 \subseteq \mathbb R^2$ defined by $t \mapsto (\cos t, \sin t)$. It is clear that $f$ is continuous and bijective. It is not open and therefore not a ...
2
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51 views

Is $f$ continuous if for every $p$, there is a sequence $p_n \to p$ such that $f(p_n) \to f(p)$?

Let $(X, d)$ be a metric space and $f : X \rightarrow X$ a function that satisfies the following property: For every $p \in X$ there exists a sequence $\{p_n\}\subset X$ such that $p_n \rightarrow p ...
2
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41 views

No norm consistent with given topology

Given the (Frechet) topology on the Schwartz class $S(\mathbb{R}^d)$ induced by the seminorms $\rho_{\alpha \beta}f = \operatorname{sup}_{x \in \mathbb{R}^d}|x^{\alpha}\partial^{\beta}f|$, how can I ...
2
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46 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
2
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66 views

Criterion for orientability: Derivative of transition map

The definition I have been given for a smooth abstract surface, $S$, to be orientable is that given a continuous family of maps $f_t: D \to S$ that embed the closed unit disk into $S$ with $f_0(D) = ...
2
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46 views

Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
2
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41 views

Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...
2
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98 views

Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and $\mathcal{O}_K^...
2
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35 views

Is the map that builds the map into the pullback continous with the compact-open topology?

I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map $$Top(T,P) \rightarrow Top ...
2
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63 views

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 ...
2
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36 views

$A$ 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 ...
2
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0answers
48 views

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 $R^{n^2}$, ...
2
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28 views

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....
2
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56 views

Homemorphism between fundamental polygon and $ℝP^{2}$

I'm trying to show that the square where the opposite edges are glued together in the inverse direction is homemorphic to the real projective plane $ℝP^{2}$. I proofed that $ℝP^{2}$ is homemorphic to $...
2
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43 views

Question on connectedness and components

We know that any connected subset $C$ of $\mathbb{R}$ is an interval, so that if $C$ is bounded, then $C$ must be of one of the following 5 types: $(a,b),(a,b],[a,b),[a,b]$ with $a < b$, and $[a,a] ...
2
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0answers
67 views

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?
2
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53 views

Convergence in the space $C([a,b],M_1(\mathbb R))$.

Let $M_1(\mathbb R)$ be the space of probability measures on $\mathbb R$ with the weak(-*-)topology: $\mu_n \rightarrow \mu$ iff $\int f(x) \mu_n(dx) \rightarrow \int f(x) \mu(dx)$ for all $f \in C_b(...
2
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0answers
43 views

Accumulation Point Proof Topology Help

Show that the following definitions are equivalent (both ways). Def 1: $x$ is an accumulation point of $S$ if for all $\delta > 0$, $[(x - \delta , x) (x , x + \delta)] \cap S \not = \emptyset$ ...
2
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48 views

The plane minus a countable set homeomorphic to the plane minus an uncountable set?

Is it possible that $\Bbb R^2-C$ can be homeomorphic to $\Bbb R^2-U$ where $C$ is countably infinite and $U$ is uncountable? Intuitively I believe the answer is no, but I'm having difficulty showing ...
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66 views

Showing that the image of a polynomial map is not closed

Let $f : \mathbb{C}^3 \rightarrow \mathbb{C}^4$ be defined by $(s, t, u) \rightarrow (st, st^2+(1-s)u, st^3, 1-s)$, where $\mathbb{C}$ denotes the complex numbers. Then for some irreducible ...
2
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0answers
98 views

Lattice Version of Stone-Weierstrass

I've been reviewing Stone-Weierstrass theoerem. While reading the wikipedia page I read the following version of the theorem: Suppose $X$ is a compact Hausdorff space with at least two points and $L$ ...
2
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45 views

finite simplicial complex compact

Let $K=(V,\Sigma)$ be a finite simplicial complex. I want to show that $|K|$ is compact. I know that $K$ is a sub-simplicial complex of $\Delta^V$ with $|\Delta^V|$ compact. So I think I should show ...
2
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29 views

A question about the dimension of topological products

For each positive integer n, is the (small inductive) dimension of the topological product of n copies of the "long line", always equal to n? I ask because the "long line" is not a separable metric ...
2
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0answers
59 views

Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
2
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0answers
28 views

Bounded uniform space

I studied that we do have a concept of total boundedness in a uniform space. I was thinking whether we have a concept of boundedness also in a uniform space (that need not be a metric space). Can ...
2
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0answers
59 views

Topology Bases and Real Numbers

Let $(X,\tau)$ be a topological space. Suppose that $\mathcal{C}$ is a subset of $\tau$, and for every $U$ in $\tau$ and every $x$ in $U$ there exists a $C$ in $\mathcal{C}$ such that $x \in C \...
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62 views

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 $. ...
2
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60 views

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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244 views

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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0answers
57 views

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 ...
2
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0answers
119 views

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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95 views

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 ...
2
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0answers
41 views

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 ...
2
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0answers
79 views

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 ...
2
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82 views

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 ...
2
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0answers
754 views

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 ...