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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5answers
171 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 $\mathbb N$? Is it because it's an open interval? Can someone please explain this to me? Also ...
2
votes
1answer
57 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 ...
1
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1answer
37 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 ...
0
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0answers
40 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 ...
0
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0answers
18 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 $\Gamma$. ...
1
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2answers
47 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 ...
2
votes
1answer
77 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?
0
votes
0answers
33 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 ...
1
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1answer
43 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 ...
1
vote
1answer
53 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 ...
2
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0answers
37 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
votes
1answer
39 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 ...
1
vote
1answer
38 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 ...
1
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0answers
34 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 ...
1
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0answers
32 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.
0
votes
0answers
60 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 $(G,\mathcal T)$ ...
0
votes
3answers
68 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 ...
0
votes
2answers
81 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 ...
0
votes
2answers
46 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.
0
votes
1answer
48 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 ...
0
votes
1answer
50 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 ...
0
votes
1answer
29 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?
1
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0answers
28 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 ...
-3
votes
1answer
91 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 ...
1
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0answers
42 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?
1
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0answers
62 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 ...
2
votes
1answer
45 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 ...
1
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2answers
30 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 ...
0
votes
1answer
34 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, ...
0
votes
1answer
25 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 ...
0
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1answer
29 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 \} = ...
-4
votes
2answers
75 views

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

$ 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.
1
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2answers
40 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?
0
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2answers
51 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 ...
1
vote
1answer
31 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 ...
2
votes
0answers
36 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 ...
1
vote
1answer
80 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\ ...
1
vote
2answers
44 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 $X$ ...
0
votes
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?
0
votes
2answers
45 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 ...
0
votes
1answer
40 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 ...
1
vote
1answer
56 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 ...
2
votes
6answers
96 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 ...
1
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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 ...
1
vote
1answer
61 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 $\mathbb{R}^\mathbb{R}$?
0
votes
1answer
85 views

Properties of preimages and intersections of sets

I am working through Bert Mendelson's "Introduction to Topology" and am having some trouble with proofs. The text in well presented but to get a proper understanding I am working through the ...
3
votes
2answers
42 views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
2
votes
3answers
76 views

Openness of path connected components of open subsets of $\mathbb C$

Let $\Omega\subset \Bbb{C}$ be an open set. My textbook states that every path connected component of $\Omega$ is open. I can't seem to understand why that is. Why does every point have to contained ...
0
votes
1answer
18 views

Strong approximation of operators.

If I want to approximate strongly an operator $T$ with another in a subset $A \in L(H)$ why is not enough to ask "for every $\epsilon>0$ there is an operator $S\in A$ such that for every $\eta \in ...
0
votes
1answer
40 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...