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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1answer
34 views

A metric on the set of closed bounded subsets of a metric space

Define the distance from a point $p$ in a metric space $(X,d)$ to a subset $Y \subset X$ by $$d(p,Y) := \inf \{ d(p,y) : y \in Y \}.$$ For any $\varepsilon > 0$, define $$Y_\varepsilon := \{ x ...
1
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0answers
24 views

Extending a locally isomorphic map to a continuous homomorphism

Hello. I tried to prove this theorem. I managed to extend it to a homomorphism. But when I tried to prove continuity (enough to show it is continuous in 0), I could not do it. I tried to take a ...
0
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1answer
12 views

Codimension of the image of the polynomials subspace is infinite

Consider the interval $I=[0,1]$ and the Banach space $E$ of real continuous functions defined on $I$ ($E=\mathcal C_{\mathbb R}(I))$. $P \subset E$ is the subspace of polynomial functions (restricted ...
2
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2answers
26 views

An open set containing a closed but not compact subset of a metric space need not contain any $\epsilon$-neighborhood of the closed set

This question concerns exercise 2(e) from section 27 (p.177) in Munkres' Topology: Let $(X,d)$ be a metric space, and let $A$ be a non-empty subset of $X$. For any point $x \in X$, we ...
2
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1answer
45 views

Name for a continuous surjection such that $\operatorname{cl}(f^{-1}(A)) = f^{-1}(A') \implies \operatorname{cl}(A) = A'$

Consider a continuous surjective map $f \colon (X, \tau) \to (X', \tau')$ satisfying $$\operatorname{cl}(f^{-1}(A)) = f^{-1}(A') \implies \operatorname{cl}(A) = A'$$ for all $A, A' \in \wp(X')$ I ...
1
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1answer
54 views

How to negate: not a limit point (symbolic logic)

1.There are few I have seen here. $\forall N(x), \exists x'\in B, (x'\neq x\wedge x'\in E)$. $\forall N(x), \exists x'\in B, (x'\neq x\to x\not\in E)$. $\forall r>0, \exists x'\in N_r(x)\cap E, ...
0
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2answers
26 views

How to prove that E's limit point must be in E? (rudin)

How to prove that E's limit point must be in E? Thm 2.23 E's open iff E_c is closed. First, suppose E_c is closed. Choose x belongs to E. Then x doesn't belongs to E_c, and x is not a limit point ...
2
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1answer
35 views

Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
1
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1answer
63 views

Is the product topology the most finest topology you can give to the cartesian product and why?

I was reading about box and product topology which are given to Cartesian products . I want to know that is the product topology(excluding the box topology) the finest topology that I can give to a ...
2
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1answer
53 views

Homeomorphism between $S^2$ and $CP^1$ via uniqueness of quotient

I am trying to show that $S^2$ and $\mathbb{C}P^1$ are homeomorphic making use of the following result - see e.g. Jack Lee Introduction to topological manifolds. Let $Y \xrightarrow{\pi_1} X_1 $, $Y ...
2
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2answers
55 views

Borel set of $\mathbb R^n$ with $n > 1$

According to various sources, the Borel set over $\mathbb{R}^n$ can be defined in several equivalent ways: For instance, it can be defined as the smallest sigma-algebra containing every open set of ...
0
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0answers
51 views

proof with 3 quantifiers? [on hold]

Problems: $$∃x ∈ S \text{ s.t. } ∀y ∈ S, ∀z ∈ S, \text{ if } z > y, \text{ then } z ≥ x + y.$$ $$∀x ∈ S, ∃y ∈ S \text{ s.t. } ∀z ∈ S, \text{ if } z > y, \text{ then } z ≥ x + y.$$ What I have ...
0
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0answers
61 views

How to write a “set is open” or “set is closed” in a pure symbolic way with quantifiers? (FOL)

How to write a "set is open" or "set is closed" or "a set is open" in a pure symbolic way with quantifiers? And how to use pure symbol to prove "E is open iff its complement is closed"? and that ...
0
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1answer
25 views

support compact modulo subgroup

I am studying (co)-induced representations of topological groups and I came across the following situation: $G$ is a topological group, $H$ a closed subgroup and $f\colon G\to W$ a set-theoretic map, ...
3
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1answer
23 views

Lower Limit Topology Properties

I am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of ...
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0answers
24 views

Complex Projective Space as a Quotient of a Disc

I am reading Hatcher's book and I have a problem understading how the complex projective space $\mathbb CP^n$ can be realised as a quotient of $D^{2n}$ (page 7) Let me briefly outline his arguments ...
1
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1answer
29 views

Prove (in the example) that being homotopic depends on the range of the Homotopy

Question: Define $F : [0,1]\times [0,1] \rightarrow X$ by $F(x, t) = (cos(\pi x), (1 - 2t) sin(\pi x))$. Take a straight-line homotopy between $F(x, 0)$ and $F(x, 1)$. Show that they are ...
0
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0answers
40 views

Is the unit square a $2$-manifold in $\mathbb{R}^2$?

I'm using the following definition of a (smooth) manifold: It's from J.Munkres "Analysis on Manifolds". This is an exercise taken from this book: Is the unit square $[0,1]\times [0,1]$ a ...
0
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2answers
58 views

Open ball metric space vs open set topological space

I'm having trouble understanding the notion of an open set when applied to a space without a metric defined on it - I have read that all metric spaces are naturally a topological space, but the ...
-1
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2answers
47 views

Is this sequence is dense?

Define $S _m, _n = $ n th smallest square number which is bigger or same than $10^ {m-1}$and smaller than $10^m$ Then is the sequence $ \frac{S_m,_n} {10^m}$ is dense in (0,1) or arbitary ...
2
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0answers
34 views

Prob. 2 (a), Sec. 26 in Munkres' TOPOLOGY, 2nd ed: Is every set compact in the finite-complement topology?

Here's Prob. 2 (a), Sec. 26 in Topology by James R. Munkres, 2nd edition: Show that in the finite-complement topology on $\mathbb{R}$, every subspace is compact. I think I can show this. Now ...
2
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1answer
44 views

Why is a simply connected 3-manifold a homotopy 3-sphere?

I recently looked at the statement of the Poincare conjecture, and realized I didn't know why the fact that a 3-manifold is simply connected implies that it is homotopic to a 3-sphere. Could someone ...
0
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1answer
28 views

$f$ proper but not universally closed

Say that a continuous function $f$ is universally closed if $f \times 1_T$ is closed for all topological spaces $T$, and call a function proper if inverse images of compact sets are compact. I know ...
1
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1answer
23 views

How to prove that a Banach space of analytic functions containing $H^\infty$ except the origin is simply connected?

If $X$ is a Banach space of analytic functions on the unit disk $D$ which contains the space of analytic bounded functions on $D$, how can I prove that $X\setminus\{0\}$ is simply connected?
4
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1answer
52 views

Munkres, Chapter 2, question on locally finite family of sets

I've been working through the Munkres Topology text on my own, and I am not sure if the following argument is correct. Fishing around the internet a bit for some alternative answers and it looks like ...
0
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1answer
35 views

Closure of an interval in specific topology

Given a topology constructed on elements of the type $$ (p,q] $$ where $p$ and $q$ are rational numbers. Let us call this topology $\tau_{-,Q}$, given an interval of type $(a,b]$ its closure in the ...
0
votes
1answer
21 views

Diffeomorphism between covering spaces

Let $\pi_1: M \rightarrow M_1$ and $\pi_2: N \rightarrow M_2$ be two smooth covering maps. Now $\phi: M \rightarrow N$ is a smooth diffeomorphism. Does this induce a smooth diffeomorphism $f: M_1 ...
4
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5answers
61 views

Explanation of $\overline{\lim} A_n$ and $\underline{\lim}A_n$

Let $(A_n)_n$ be a countable family of subsets of a set $X$. We define: $$\lim \inf A_n = \underline{\lim} A_n = \bigcup_{n \in \mathbb N} \bigcap_{k \ge n} A_k$$ $$\lim \sup A_n = \overline{\lim} ...
1
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1answer
31 views

Homeomorphism of the closed unit ball not preserving the sphere?

Exercise 2.9.12 in Ronnie Brown's Categories and Groupoids asks the reader to show that if $f:\mathbb{R}^n \to \mathbb{R}^n$ is continuous such that $f$ restricts to a homeomorphism from the open ...
0
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1answer
49 views

how shall i'll prove if c={(x_n) :exists lim x_n}is a Hyperplane, dense, or closed? [on hold]

Let $$c=\{(x_n) :\exists ~ \lim x_n\},$$ where $c$ is included in $\ell^\infty$. How can I find a function $T$ such that $\ker(T)=c$? Also, after that, how can I see if $T$ is continuous? Thanks.
4
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0answers
55 views

Properties of $\mathbb{C}P^n$

I'm currently working on a somewhat deformed version of $\mathbb{C}P^2$ and want to check some properties from a geometrical and/or topological point of view. Of course, $\mathbb{C}P^2$ is Kähler ...
1
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1answer
14 views

Open condition given by inequality on functions

Let's say we have two functions $f,g\in C^\infty(D)$, $D$ an open domain in $\mathbb{R}^2$. The condition $f(x,y)<g(x,y)$ is an open condition on $D$? With this I mean: do the points $(x,y)\in D$ ...
3
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0answers
49 views
+50

Cantor Set in Alexander Horned Sphere Construction

I have seen it said in several different places that in the standard construction of the Alexander horned sphere, given by successive embeddings of a sphere with $2^n$ handles, either limited or ...
1
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1answer
84 views

why separable normal space has only continuum many different open subsets?

I cannot prove the fact in the title. Please help! I am reading the handbook of set theoretic topology. And I found this fact in a proof in the book. Thank you.
0
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0answers
33 views

Connectedness of circle without center line across it

Using a definition I saw in an old Russian book, a set in $\mathcal R^{n}$ is said to be connected if it cannot be represented as a disjoint union of two nonempty, separated sets. Separated, meaning ...
1
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3answers
44 views

How to determine the closure of a subset and prove it is actually the closure?

I have this subset $E = \{r \in Q: r^2 \leq 2\}$ which is in $\mathbb{R}$ with the Euclidian metric. I was wondering how can I find the closure of this subset. Here is what I have: The limit points ...
2
votes
2answers
34 views

Kernel of a bounded linear operator on a normed linear space need not be closed or open?

How should be the kernel of a bounded linear operator on a normed linear space as a set? Kernel of a bounded linear operator on a normed linear space need to be closed or open? Or it need not be ...
1
vote
1answer
76 views

Finite mapping $f : \mathbb R^2 \to \mathbb R$

Is there an continous function $f: \mathbb R^2 \to \mathbb R$ such that $f^{-1}(a)$ is finite for every $a \in \mathbb R$? It's not possible for analytic or smooth but I'm curious about continous ...
17
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2answers
301 views

Completion of the real numbers

On the real line $\mathbb{R}$ endowed with euclidean topology i may put different metrics, inducing the same topology, but inducing different completions. For example if one considers the standard ...
3
votes
1answer
45 views

Surjectivity of $\mathcal{id}_{\mathbb{R}^n}+g$ when $g$ is a contraction?

Assume $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is a contraction and consider $h=\mathcal{id}_{\mathbb{R}^n}+g$. The map $h$ is injective. Is it always surjective? My question has the following ...
0
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1answer
48 views

What does arbitrary mean in the following context?

I recently encountered a theorem stating, "The arbitrary product of compact spaces is compact". What does arbitrary product mean in this context? Any product of compact spaces? If that is the case why ...
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0answers
79 views

Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
1
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0answers
23 views

What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
2
votes
1answer
102 views

Prove that $x+g$ is homeomorphism

Problem: Assume we have $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ of $C^1$ class with derivative bounded uniformly by some constant $M<1$. Consider ...
1
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0answers
15 views

Proposed proof for quasi-metric result

A quasi-metric on a set $X$ is mapping $\rho: X \times X \rightarrow [0, \infty)$ satisfying the following conditions: $\rho(x,y) \geq 0~~\text{and}~~\rho(x,x) = 0;$ $\rho(x,z) \leq \rho(x,y) + ...
3
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1answer
34 views

How do I prove the converse of Stone-Weierstrass theorem?

Let $X$ be a locally compact Hausdorff space. Let $\bar \rho$ be the uniform metric on $\mathbb{R}^X$ and $\mathscr{A}$ be an $\mathbb{R}$-subalgebra of $C_0(X,\mathbb{R})$ which is dense in ...
0
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1answer
29 views

Outer Regularity of the Lebesgue measure on the Hilbert brick

Is the product measure on the Hilbert brick $I=[0,1]^\mathbb{N}$ outer regular (that is measure of every set is the inf of measures of open sets, containing it)?
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5answers
59 views

Finite set of points of $R^n$ is compact

In order to show that a finite set of points of $R^n$ is compact, I just need to show that the set is closed and bounded. First of all, since it's a finite set, I can Always pick the greatest ...
3
votes
1answer
42 views

Is this space Hausdorff and are these two spaces homeomorphic?

Let $S^1 = \{(x,y) \in \mathbb{R^2} \ | \ x^2 + y^2 = 1 \}$. We define an equivalence relation $\sim$ on $S^1$ such that $(x,y) \sim (x',y')$ if and only if $y = y'$. Now we study the following ...
0
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0answers
28 views

Cartesian product between manifolds

I was given the following exercise: Show that if $M$ is a $k$-manifold without boundary in $\mathbb{R}^m$, and if $N$ is an $l$-manifold in $\mathbb{R}^n$, then $M \times N$ is a $k+l$ manifold in ...