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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3
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2answers
63 views

Characterising the discrete topology with compact subsets [duplicate]

If a set is endowed with the discrete topology then a subset is compact iff it is finite. Is the converse true? That is, given a Hausdorff topological space such that every compact subset is finite, ...
0
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1answer
28 views

Two proofs with possibly Baire category theorem about completness.

I'm working with completness right now and I've come across two interesting problems. In my opinion they are worth a little bit attention . a) Let $K$ be closed subset with empty interior on ...
2
votes
1answer
85 views

Decomposition of open sets in $\mathbb{R^d}$

I am trying to prove the following problem. It's an exercise in Stein's Real Analysis text book. Problem: Suppose $\mathbb{R^d}-\{0\}$ is represented as $\mathbb{R_+}\times S^{d-1}$ with ...
1
vote
1answer
32 views

$f:X\to Y \text{ is continuous} \iff f^{-1}(A^*) \subseteq (f^{-1}(A))^*$

Really struggling with exercise 9.10 from Sutherland's "Introduction to Metric and Topological Spaces". Any help would be greatly appreciated. Let $(X,t), (Y,t)$ be topological spaces, and $f: X \to ...
0
votes
1answer
45 views

The closed and bounded sets are compact in the product topology

Let $X=\mathbb{R}^{\aleph_0}$ with the product topology, it is true that all the closed and bounded (in the uniform sense) sets are compact?
1
vote
1answer
42 views

Continuity of function and topology

I have this exercice $E=\{a,b,c,d\}$ with the topology $\tau=\{\emptyset, \{a\},\{a,b\},\{a,b,c\},E\},$ and the space $F=\{x,y,z,w\}$ with the topology $\theta=\{\emptyset.\{y\},\{y,z,w\},F\}$ I ...
1
vote
0answers
16 views

path connectedness of space of almost commuting matrices

Let $R$ be a topological ring which is a domain. Let $n$ be an integer and let $\zeta_n$ be a $n$-th root of unity. Denote by $X$ the set of $m$ by $m$ invertible matrices with coefficients in $R$ ...
2
votes
3answers
48 views

Show that the sequence of functions $(x_n)_{n≥1}$ in $C[0, 1]$ given by $x_n(t) = t^{2n} − t^{3n} , ∀t ∈ [0, 1]$ is bounded

That is $C[0,1]$ equipped with the supremum metric. I have proven, using derivatives, that each function $x_n$ has a local maximum and local minimum at $(2/3)^{1/n}$ and $0$ respectively. I know ...
7
votes
0answers
152 views

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
3
votes
1answer
58 views

Hausdorff dimension of $\lim_{n\to\infty}\sin(2^nx)$

Calculate the Hausdorff dimension,$\dim_H$ of $$S=\{x\in(0,1):\lim_{n\to \infty}\sin2^nx=0\}$$ By definition We need to find the minimal $\alpha$ s.t $\sum_{i\in I}|U_i|^\alpha$ is minimal where ...
2
votes
1answer
40 views

Example of topological space where there is a point and a subset $A $: $x \in \overline A $, but no sequence in $A $ converging to $x $?

It is known that if a space $X $ is metricable then for any subset $A $ of $X $ and a point $x \in \overline A $ there is a sequence of points in $A $ converging to $x $. I wonder if there is an ...
0
votes
1answer
40 views

Convergent squence in topology

Please, I consider this topological space $(E,\tau)$ where $\tau=\{G\subset E, ~\text{card}~ (E\setminus G)~\text{countable}\}\cup\{\emptyset\}$ How to prove that a sequence $(x_n)$ is convergent in ...
0
votes
2answers
21 views

Show that τ = {A ⊂ R : ∃N ∈ N, ∀n ≥ N, 1/n ∈ A} ∪ {∅} is a topology for R. n

To show that τ is a topology for R, we have to show that the empty set and R are open. We also have to show that intersection of two open sets is open and that the union of open sets is open. I am ...
0
votes
1answer
60 views

Is there a topological proof that additon and multiplication are continous functions from $\mathbb R \times \mathbb R $ into $\mathbb R $?

Is there a topological proof that additon and multiplication are continous functions from $\mathbb R \times \mathbb R $ into $\mathbb R $? That is, can we prove continuity using the topological ...
1
vote
0answers
133 views

Line with two origins is a manifold but not Hausdorff

The line with two origins is $(\mathbb{R} \times \{0,1\})/\sim$ where $(x,0)\sim(x,1)$ for $x\neq 0$. I can see that it is not Hausdorff, since we cannot separate the points $(0,0)$ and $(0,1)$. ...
1
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0answers
101 views

Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
2
votes
2answers
105 views

Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not ...
3
votes
1answer
68 views

Cut Möbius Band

$$\text{Cut a Möbius band from its center line, and then what do we get?}$$ Someone may find it's not easy to imagine without a paper in hand. However, if we cut a square paper from center line at ...
0
votes
2answers
62 views

Group with topology which is not topological group

What will example of a group $G$ with topology $\tau$ such that $f: G \to G$ such that $f(x) =x^{-1}$ and $g: G \times G \to G$ such that $g((x,y)) = x * y$ (where $*$ is binary operation on $G$) both ...
0
votes
1answer
50 views

a simple question about the density convergence of sequences

Definition 1: A sequence $ \{x_n, n=1,2,3,...\}$ of points in a topological space $X$ converges to a point $x\in X$ in density if for any neighborhood $ V$ of $x$ in $X$, $x_n\in V$ but for a set ...
1
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0answers
47 views

Condition ici=ic on a topological space is equivalent to if each dense set has dense interior in the space.

I am required to prove the following: Let $(X,\tau)$ be a topological space.Then each dense set has dense interior iff $ici=ic$ holds where $i$ is the interior operator and $c$ is the closure ...
5
votes
1answer
58 views

Definiton of No Tear and No Paste

Topologists often mention an example beginning by "If there is no tear and no paste, then ...". As a student, I am confused with this "term", and I want to know the exact mean of it. First of all, ...
0
votes
2answers
31 views

Countable Neighborhoods

How to prove it? Let $X$ be a topological space with base $B$ and $x \in X$. Show that, when $x$ has countable neighborhood basis, so there is a countable subset of $B$, which is a neighborhood basis ...
0
votes
1answer
74 views

continuous poset w.r.t. Scott topology

I am learning continuous poset by myself. I have conclusion as follows: If $P$ is a continuous poset w.r.t. Scott topology then there is $x\in P$ s.t. for any $y\in P$ and for any open sets $U_x$ and ...
4
votes
1answer
108 views

Automorphism group of a topological space

Let $G$ be any group. Is there a topological space $(X,\tau)$ such that the automorphism group $\textrm{Aut}(X,\tau)$ is isomorphic to $G$?
3
votes
2answers
86 views

$\Bbb{R}^2$ not homeomorphic to $\Bbb{R}^2\setminus \{0\}$ [duplicate]

I would like to show that $\Bbb{R}^2$ and $\Bbb{R}^2\setminus \{0\}$ are not homeomorphic without using Algebraic Topology. Is there an elementary way to do this?
3
votes
2answers
85 views

Are there any disadvantages to working in the category of k-spaces as opposed to Top?

Unlike the category Top of topological spaces with continuous maps as the arrows, the full subcategory of compactly generated spaces (k-spaces) is Cartesian closed. It seems like a very nice ...
1
vote
2answers
79 views

A closed set $F \subset \mathbb{R}$ such that $F, F', F'', F''',\dots $ are all distinct

Let $F \subset \mathbb{R}$ be a closed set. Let $F'$ be the set of the limit points of $F$. Question: Does there exists a set $F$ such that $F, F', F'', F''', ..... $ are all distinct and nonempty? ...
0
votes
2answers
78 views

Showing that a countable product of unit intervals is not compact in the box and uniform topology.

Let $I = [0,1]$ and $I^{\omega}$ be the countable product of unit closed interval I, where each $I$ is given the subspace topology of $R$ in the usual topology. I am trying to show that $I^{\omega}$ ...
0
votes
1answer
86 views

Is the following set path connected? [duplicate]

Let $A$ be the set of all those points of plane $\mathbb{R} ^2$ in which both coordinates are rational or both are irrational. Is $A$ path connected?
3
votes
3answers
290 views

Can a Set Have Infinitely-Many Non-Homeomorphic Topologies?

Let X be a set. Is it possible for X to have an infinite number of topologies up to homeomorphism (i.e. infinitely-many different topological structures)?
1
vote
1answer
52 views

Closedness and boundedness in metrizable topological spaces

This is a quick question that I have not managed to answer myself: let $X$ be a metrizable topological space, let $A\subset X$ be a closed, bounded subset. $f:X\to Y$ is a homeomorphism, must $f(A)$ ...
0
votes
1answer
38 views

Show that the cylinder is not ambient isotopic to the Mobius band.

Here is my definition for ambient isotopy: We say if there is an orientation preserving piecewise linear homeomorphism $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ (or replace $\mathbb{R}^3$ with $S^3$) ...
1
vote
1answer
43 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
4
votes
0answers
216 views

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
2
votes
4answers
82 views

Show the subset $A$ of $\mathbb{R}^n$ is compact

Show the subset $$A = \{(x_1, . . . , x_n) ∈ \mathbb{R}^n| −1 ≤ x_1 ≤ x_2 ≤ · · · ≤ x_n ≤ 1\} \subset \mathbb{R}^n $$ is compact, and show the function $$\left\{\begin{array}{}f : A → ...
1
vote
0answers
34 views

Proof of Triangulation Theorem for 1-Manifolds

While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102. Note that Int$\ e\cap\ $Int$\ e'$ is open in ...
0
votes
1answer
39 views

How can I show that if a set is bounded, then it's contained in a k-cell?

The set is a bounded subset of R (under the Euclidean metric), and a k-cell is a set of points {x_1...x_k} such that a_j < x_j < b_j for j=1...k. Any ideas on how to show this?
2
votes
1answer
61 views

Constructing almost disjoint families

Let $\mathcal A$ be an almost disjoint family of subsets of $\omega$ and let $\Psi (\mathcal A)$ be the Mrówka space (definition here). Let $$\mathcal I (\mathcal A)=\{X\subseteq \omega : X\subseteq ...
2
votes
1answer
123 views

Torus/Möbius Band homeomorphism

Is a fattened Möbius Spiral Band homeomorphic to a Torus? (Due to the same Euler Characteristic $\chi$ ?) Are both non-orientable? Following (3D printable plastic) Torus has a square section that ...
0
votes
3answers
78 views

Boundary of $\mathbb{R}^4$ and fundamental group of $\mathbb{R}^4/\mathbb{R}^2$

a) To what I know boundary makes no sense in open sets. Does it make any sense to talk about the boundary of $\mathbb{R}^4$? In physics they consider it when discussing wether the universe has a ...
2
votes
1answer
41 views

Fixed point of a mapping

How to prove that every continuous $f:S^1 \to S^1$ such that $deg(f)\neq 1$ has a fixed point? One hint is that if $f(x)\neq x$ for any $x\in S^1$ then $f$ is homotopic to the antipodal map $a$ but I ...
1
vote
2answers
48 views

Short proof using continuity and set conclusion

I'm new to uni math and in my most recent assignment I got stuck trying to proof the following: Let $a,b \in \mathbb{R}$ and $a<b$. Suppose $\space f:[a,b] \rightarrow \mathbb{R}$ be continuous. ...
0
votes
3answers
74 views

Path connectedness of the set of points $(x,y)$ where $x$ is rational or $y$ is rational [duplicate]

Prove that $X=\{(x,y) :x\text{ is rational or }y\text{ is rational}\}$ is path connected. So for every $(x,y)$ in $X$, I need to find a continuous function $f$ on $[a,b]$ such that $f(a)=x$ and ...
0
votes
3answers
49 views

Determine whether the set $X=\{(a,b) : |b|>e^a \}\subset \mathbb R^2$ is connected

Determine whether the set (as a subspace of $\mathbb R^2$) is connected. $$X=\{(a,b) : |b|>e^a \}$$ Thoughts: Not sure how to go about this question. I suppose look for a partition. Anyone got ...
0
votes
1answer
29 views

Baire category theorem in use on a plane

Let $F\subset\mathbb{R}$ be a closed nowhere dense set. One must show there exists $(a,b)\in S^1$ for which $b\neq qa+c$, for all $q\in\mathbb{Q},c\in F$. It's my second question concerning Baire ...
1
vote
0answers
33 views

Is a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable; homeomorphic to space of rationals?

Let $X$ be a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable. Is $X$ homeomorphic to the space of rationals? $X$ is called nowhere compact when ...
1
vote
1answer
91 views

Compute the map $H^*(CP^n; \mathbb{Z}) \rightarrow H^*(CP^n, \mathbb{Z})$

I'm trying to solve problem 3.2.6 in Hatcher. The problem is stated: Use cup products to compute the map $H^*(CP^n; \mathbb{Z}) \rightarrow H^*(CP^n, \mathbb{Z})$ induced by the map $CP^n ...
1
vote
0answers
32 views

Is $S(\mathbb{R}^{d})$ dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$?

Let $S(\mathbb{R}^{d})$ denote the class of Schwartz functions in $\mathbb{R}^{d}$. Is it true that $S(\mathbb{R}^{d})$ is dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$, the locally integrable ...
1
vote
0answers
32 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 ...