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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Path- connected

Let $a=(a_1,\dots, a_k)$ and $b=(b_1,\dots, b_k)$ be points in $k$-dimensional space $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function on the unit interval $[0;1]$ with values in ...
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0answers
12 views

Volterra operator and completely continuous operators

Consider the Volterra operator $V$ defined here. Let me give some definitions first: [Dunford-Pettis] We say that a bounded linear operator $D:L_1[0,1]\to L_1[0,1]$ is Dunford-Pettis if it sends ...
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5answers
92 views

Is $\mathbb N$ dense in $\mathbb R$?

Is $\mathbb N$ dense in $\mathbb R$? Let say (a,b)=(0,1) How come it does not contain an element of ℕ ? Is it because it's an open interval? Can someone please explain this to me? Also when would ...
2
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1answer
34 views

Cardinal characteristics

Assuming Continuum hypothesis is not true, How many cardinals $k$ exist which are $\aleph_1 < k < \mathfrak c$? Can I assume that there is a finite number of these cardinals or is there an ...
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1answer
35 views

Contradiction in an Alternative Definition of an Open Set?

A set $G$ in $\mathbb R^p$ is said to be open in $\mathbb R^p$ if , $\forall x \in G$, $\exists r \in \mathbb R^+$ such that every point $y$ in $\mathbb R^p$ satisfying $|x-y|<r$ also belongs to ...
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0answers
32 views

Error? An open subset of $\mathbb R^p$ is connected if and only if it can be expressed as the union of two disjoint non-empty open sets.

I believe the book which I am reading has a printing error. One of the lemmas reads like this An open subset of $\mathbb R^p$ is connected $\iff$if it can be expressed as the union of two disjoint ...
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0answers
21 views

Boundary, closure, interior [on hold]

$X=(0,4] \cup \{6\} \cup[10,11]$ is subspace of $\mathbf{R}$. If A is $A=(0,2] \cup \{6\} \cup(10,11]$, find $IntA$, $ClA$, $FrA$ in subspace $X$, where is: Int - interior, Cl -closure, Fr-boundary. ...
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0answers
13 views

Natural surjection from complex upper half plane into modular curve

I am considering the natural surjection $\pi : \mathcal{H} \to Y(\Gamma)$ where $\mathcal{H}$ is the complex upper half plane and $Y(\Gamma)$ the modular curve of the congruence subgroup $\Gamma$. ...
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2answers
41 views

It's the discrete topology.

I have to proof that if I have $(X,\tau)$ and $(Y,\delta)$ two topological spaces, if every function $f:X\longrightarrow Y$ is continuos then $\tau$ is the discrete topology. I don't know what is the ...
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1answer
72 views

Are $\Bbb R^2$ and $\Bbb R^3$ homeomorphic?

I know $\Bbb R$ and $\Bbb R^2$ are not homeomorphic.but Are $\Bbb R^2$ and $\Bbb R^3$ homeomorphic?
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0answers
26 views

Disconnected Sets definition and connectedness of the unit interval

The definition of a disconnected set seems a bit ambiguous in the book I am reading : $1.$ A subset $D$ of $\mathbb R^p$ is said to be disconnected if there exist two open sets $A$ and $B$ such ...
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1answer
41 views

subspaces of $\mathbb{R}^2$ that are not retracts of $\mathbb{R}^2$.

I have to either prove or disprove: There exist infinitely many subspaces (up to homeomorphism) of $\mathbb{R}^2$ that are not retracts of $\mathbb{R}^2$. I can think of a few subspaces (e.g., finite ...
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1answer
42 views

Irreducible components of fiber bundle

Suppose $\pi:X \rightarrow Y$ is a (locally trivial) fiber bundle $F$, where all spaces are Noetherian. Suppose $F$ and $Y$ are irreducible; show that $X$ is also irreducible. Here a similar ...
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0answers
33 views

Riesz-Markov-Kakutani Theorem: Various Versions

The Riesz-Markov-Kakutani theorem usually comes in various versions. So I'm a little bit confused and wondering which of these are right. Let $\Omega$ be a locally compact space. Then: Complex ...
2
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1answer
35 views

Three-space property

I have found two definitions of a three-space property. One definition is: $(P)$ is a three-space property if whenever $E$ Banach space, $F\subseteq E$ is a closed linear subspace and two of the ...
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1answer
32 views

complement of zero set of holomorphic function is connected

I'm stuck with the following part of exercise 1.1.8 in Hubrechts book Complex geometry: Prove that, if $U \subset \mathbb C^n$ is open connected, then $U \setminus Z(f)$, the complement of zero set ...
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0answers
32 views

to prove that a metric space which is not complete [on hold]

Given $C_{\infty}= \{x=(x_n) :x_n \in \mathbb{R}\ \text{and}\  \exists\ n(x)\in \mathbb{N} $ s.t $x_n = 0\ \forall\ n > n(x)\}$  Where each element is sequence of type ...
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0answers
26 views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
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0answers
28 views

Problem about compact subspace of Hilbert cube.

This is my problem: I have already completed part (i), but I really can't see how I can relate compact subspace with homeomorphism in part (ii). Please give me some ideas.
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0answers
17 views

Is a partially topological group completely regular

Let $G$ be a group and $\mathcal T$ be a topology on $G$ and the function $$ \begin{align*} &f:G\times G\to G\\ &f(x,y)=xy^{-1} \end{align*} $$ be continuous at $(1,1)$. Is $(G,\mathcal T)$ ...
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3answers
64 views

Why we need topology to define these new form open sets?

By the definition of topology, I feel topology is just a principle to define "open sets" on a space(in other words, just a tool to expand the conception of open sets so that we can get some new forms ...
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2answers
76 views

Characterization of continuity in terms of preimages of open sets

1--8 Theorem. If $A\subset \mathbb R^n$, a function $f:A\to \mathbb R^m$ is continuous if and only if for every open set $U\subset \mathbb R^m$ there is some open set $V\subset \mathbb R^n$ such ...
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2answers
45 views

Is $\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$ compact in $\Bbb R^2$?

Is this set in $\Bbb R^2$ compact: $$\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$$ I think it is compact, but the answer says not. Any help is appreciated.
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1answer
40 views

Axioms of seperation

I am studying topological spaces, and I have seen that there are $3$ main axioms of separation: $\mathrm{T1}$, Hausdorff and normal. Now, between Hausdorff and normal there is a case where: given ...
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1answer
42 views

open set and cardinality

I've learnt Set Theory, but I didn't learn Topology and Measure Theory. I met a term "open set" today. According to Wiki, An open set is an abstract concept generalizing the idea of an open ...
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1answer
21 views

cells of quotient CW complex

Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
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0answers
32 views

topology problem read [on hold]

Not a topology since if we consider $\{x_n\}_{n=1}^{\infty}$ where $x_n = 1- \frac{\sqrt{2}}{2n}$. Clearly, $(x_n \in \mathbb{R}^+ \backslash \mathbb{Q})$, $(\forall n \in \mathbb{N})$. Let $(A_n = ...
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0answers
25 views

Diffeomorphism and hyperbolic points

Suppose $f$ is a diffeomorphism.Prove that all hyperbolic periodic points are isolated. I tried using the mean value theorem using two diferent periodic points (assuming the periodic points arent ...
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1answer
90 views

If $A\times B$ is compact , then $A$ is compact and $B$ is compact?

Is this true? I think so but I can't seem to prove it / know how to. If $A\times B$ is compact then if $(x,y) \in A\times B$ then $x \in A$ and $y\in B$ and $(x,y)$ is covered by finite subsets of a ...
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0answers
37 views

Are Borel sets preserved by an open continuous map?

Does an open, continuous function defined on a compact metric space to itself send Borel sets to Borel sets?
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0answers
55 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
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1answer
41 views

Proof of paracompactness of CW-complexes (J. Lee, Introduction to Topological Manifolds)

I have a question about a proof in John Lee's Introduction to Topological Manifolds (5.22). Given CW-complex $X$ with skeletons $X_n$ and open cover $\left(U_\alpha\right)_{\alpha\in A}$, we ...
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2answers
29 views

Homology group of 3-fold sum of projective planes

I want to calculate the homology group of the 3-fold sum of projective planes defined by the labelling scheme $aabbcc$. For this I will use the following corollary from Munkres: Corollary 75.2: Let ...
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1answer
31 views

Homology group of space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$

I have to calculate the homology group of the quotient space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$ and then determine to which of the following spaces it is homeomorphic: $S^2, ...
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1answer
23 views

Arbitrary Fundamental Group and Surfaces

someone had explained to me how to construct arbitrary space $X_G$ such that $\pi_1(X_G) \cong G$, but i don't remember the end. The idea was the following : take a presentation of the group, and ...
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1answer
24 views

The set of all limit points $A'$ of a subset of a topological space $X$ is empty if $\tau = 2^X$

Proposition: If $X$ is a topological space with $\tau = 2^X$, then $A' = \emptyset$ where $A \subset X$ I found the proof and it uses the fact that if $x \in A$, then $\{ x\} \cap A - \{x \} = ...
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2answers
63 views

I neet to prove that the set $ A:=\{ \frac {1}{n} | n \in \mathbb{N}\}\bigcup\{ 0\}$ is closed in R.

$ A:=\{ \frac {1}{n} | n \in \mathbb{N}\}\bigcup\{ 0\}$ is a closed set in $\mathbb{R}$ by the definiton. I can't use that $cl(A)=A$ iff a is a closed set.
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2answers
39 views

Closed sets in product topology

I have an assignment, I have to proof that arbitrary product of close sets is closed in the product topology, I think I have to use complements and treat with opens, what do you think?
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1answer
39 views

Space of Functions: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
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1answer
29 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
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0answers
35 views

Basic question about lifting maps to covering spaces

Any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers). The space $\tilde X_1$ is certainly ...
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1answer
78 views

Is true the boundary of compact set of $\mathbb{R}^n$ have Measure Zero?

Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \rightarrow [0, \infty[$ a measurable function. Suppose that there exist $C>0$ such that $$\int_K f dm < C,\ \forall\ K\subset\Omega,\ K\ ...
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1answer
32 views

$X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point

Im trying to show that: for $X,Y$ topological spaces $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point while $[X,Y]$ denote the set of homotopy classes of maps of $X$ ...
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2answers
66 views

How to show that every continuous function from $[0:1]$ to $[0:1]$ has a fixed point?

This exercise is from Munkres topology: Let $f:[0:1]\rightarrow [0:1]$ be a continuous function. How can we prove that there exists some point $x\in [0:1]$ such that, $f(x)=x$? Any ideas please?
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2answers
42 views

Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
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1answer
37 views

Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton $$A\sim ...
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1answer
55 views

Operators on the family of all subsets of a topological space that maybe generates a base for these family.

I will try to do at least something of my first question. Given a topological space $\langle X,\tau\rangle$, we define two operators on $2^X = \{ A : A \subseteq X \}$ as follows. For $\alpha ...
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6answers
85 views

Rudin's Topological Definition of an Open Set — Does it Disagree with the Metric Space Definition?

I wanted to share this definition of an open set, which made me uncomfortable. It comes from Rudin's Real and Complex Analysis and begins with the definition of a topology: A collection $\tau$ of ...
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1answer
30 views

Homeomorphism of a Genus-2 Surface

Does there exist a homeomorphism from a genus-2 surface, the connected sum of 2 tori, to two circles, $S^1$, intersecting at a point? Intuitively it seems that the double torus can be squeezed into ...
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1answer
60 views

topology defined on the set $\mathbb{R}^\mathbb{R}$?

What is the topology defined on the set $\mathbb{R}^\mathbb{R}$ of functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that allows us to talk about convergence of sequences in $\mathbb{R}^\mathbb{R}$?