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
33 views

$T:X \to Y$ bounded linear map and $X$ separable implies $Y$ is separable?

Let $T:X \to Y$ be a bounded linear map between Banach spaces. Suppose that $X$ is separable. Is it true that $Y$ has to be separable? I think yes, since the map is continuous it takes the ...
0
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0answers
27 views

Sum of the reciprocals topology

Let's define this topology in $\Bbb N$ (here $\Bbb N$ begins at $1$): $$K\subset\Bbb N\text{ is closed }\iff K=\Bbb N\;\text{ or }\;\sum_{n\in K}\frac1n\text{ converges}$$ I have worked some on it. ...
3
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1answer
43 views

Let $A ,B \subseteq \Bbb{R}^{k}$ and $A+B =\{a+b| a\in A, b\in B\}$then:

Let $A ,B \subseteq \Bbb{R}^{k}$ and $A+B =\{a+b| a\in A, b\in B\}$then: a)If $A,B$ be open then $A+B =\{a+b| a\in A, b\in B\}$ is open. b)If $A,B$ be connect then $A+B$ is connect? c)If $A,B$ be ...
6
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0answers
55 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
1
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1answer
42 views

$f$ is continuous on $E$ if and only if its graph is compact.

This question may be asked before under different formulation, the original problem is Chapter 4, Exercise 7 of Rudin's text: The Principles of Mathematical Analysis: Problem: If $f$ is defined on ...
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0answers
21 views

Any online videos on a course taught from Munkres?

Are there any vidoes available on the Internet --- for watching online or for download --- of any (general) topology course taught using the book Topology by James R. Munkres, 2nd ed? If so, please ...
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0answers
36 views

Show that $X \cup \mathcal{E}(X)$ ( $X$ and the ends of $X$) is a compact Hausdorff space.

Let $X$ be a connected,locally connected, locally compact and Hausdorff space. By an end $\epsilon$ we mean a function which maps a compact C subset of X to a connected component of $X-C$ such that ...
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0answers
37 views

Topology on partially ordered set

Let $(X, \leq )$ be a partially ordered set. How would you define a topology on $X$ such that the closed sets are precisely the order-closed sets? Where $B \subset X$ is order-closed if ...
5
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1answer
45 views

Show the discrete topology is the only one larger than $\tau_l$

$(X,\le)$ is a partially ordered set, we define $U_l(x)=\{y\ |\ y\le x\}$, and $\tau_l$ is the topology generated by $\{U_l(x)\}$. We want to prove that the discrete topology is the only on that's ...
4
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1answer
49 views

Every neighborhood $N_r(x)$ in $\mathbb{R}^n$ is connected

I am working on an exercise in baby Rudin (Ex 2.20 in particular) and as part of that I am trying to show that any neighborhood in a metric space is connected. I've seen several differing definitions ...
0
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1answer
27 views

Finite closed covering of a bounded set in $\mathbb{R}^n$

My Attempt: I think here I can define the diameter of $A$ as follows since it is bounded. diam $A=\sup \{|x-y|: x, y \in \mathbb{R}^n\}$ So, I can take each $r_k$ as diam $A$. Am I on the ...
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0answers
53 views

Books for Ordinals and Cardinals

I am looking for a nice introductory book to read to learn and master ordinals and cardinals. Please help me!
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0answers
30 views

Does there exist a kuratowski set which is uncountable

A subset of a topological space is called the Kuratowski set if we can get 14 different sets by applying closures and complementation successively. I want to find a set which is uncountable and is a ...
1
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1answer
29 views

topological isomorphism between a group and product of its subgroups

I have stumbled upon the following question: Let $G$ be a $\sigma$-compact, locally compact Hausdorff group with $N$ and $H$ closed normal subgroups of $G$. Also $$N\cap H= \{e\}$$ and $$G=NH .$$ Then ...
2
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0answers
30 views

$gf$ closed with compact fibers $\implies f$ closed with compact fibers

Call a continuous function $\phi: A \to B$ universally closed if $\phi \times 1_T$ is closed for every topological space $T$. Exercise 3.6.13(d) of Ronnie Brown's Topology and Groupoids asks the ...
1
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1answer
12 views

Infinite set with discrete metric

Let $X$ be an infinite set. For $p\in X$ and $q\in X$, define $d(p,q)=1-\delta_{pq}$. Prove that this is a metric. Which subsets of the resulting metric space are open? Which are closed? Which are ...
0
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1answer
13 views

The set of all interior points. Set equality

$(\overline{E})^o=E^o$. Is this equality true? I proved that inclusion $E^o\subset (\overline{E})^0$ is true. But how to prove that inclusion $(\overline{E})^0\subset E^o$ is true or false? $E^o$ ...
1
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0answers
22 views

A basis $B$ for a topological space $X$ is a ring of sets iff $X\in B$

Let $X$ be a topological space with basis $B$, and suppose that $B$ has the following properties: $B$ consists of compact and dually compact$^\ast$ subsets. For every triple $U, V, W\in B$, we have ...
2
votes
1answer
34 views

Diameter of a 10-ball in a 10-box is larger than the side length of box?

I came across this idea in a lecture on elementary topology. While it makes sense algebraically, I'm hoping someone could shed some light on the way this is possible. So you begin with a square of ...
6
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8answers
1k views

What does it even mean to say 'preserve structure'? [duplicate]

Could somebody give a concrete example of a group structure being preserved in a isomorphism, et cetera? I always hear this 'preserve structure' thing. Ok, could somebody give me a rigorous definition ...
0
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1answer
37 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 ...
0
votes
1answer
15 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
32 views

Prob. 2 (e), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Open supersets and $\epsilon$-neighborhoods of closed noncompact sets

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
48 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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2answers
68 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
27 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
votes
1answer
40 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
66 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 ...
3
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1answer
56 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
votes
2answers
56 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
53 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
67 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
votes
1answer
28 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
votes
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
30 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 ...
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0answers
44 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
62 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 ...
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votes
2answers
49 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
35 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
47 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
votes
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
25 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
votes
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
36 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
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1answer
22 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
62 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
vote
1answer
33 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
votes
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
votes
0answers
56 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 ...