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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12 views

Center of real projective line or Riemann sphere

I have recently encountered the ideas of the real projective line and the Riemann sphere, and it seems to me that in any circle (representing the real projective line) or sphere, the center is a ...
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1answer
24 views

Two nonhomeomorphic topological space may be embedded in each other. i need example pls help [on hold]

Find two nonhomeomorphic topological space X and Y such that X imbedded in Y and Y imbedded in X.
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31 views

An excerpt from a seminar

It is a statement that a professor made in a seminar which I attended yesterday.He says that the following hold: $1$.If $D$ denotes the closed unit disc then there does not exist a continuous ...
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2answers
39 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, ...
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1answer
16 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 ...
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1answer
43 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 ...
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1answer
24 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 ...
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1answer
25 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?
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1answer
35 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 ...
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11 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 $R$ ...
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3answers
33 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 ...
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18 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 ...
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1answer
51 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 ...
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1answer
33 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 ...
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1answer
34 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 ...
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2answers
26 views

Normal matrices connected?

Is the set of all normal matrices connected in $M_n(\mathbb{R})$, where the metric is the usual metric of $\mathbb{R}^{n^2}$? ($A$ is normal iff $AA^{t}=A^{t}A$.)
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20 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 ...
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1answer
51 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 ...
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0answers
21 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 $(0,1)$. ...
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0answers
31 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 ...
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2answers
54 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 ...
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1answer
47 views

Cut Mobius Band

$$\text{Cut a Mobius band from its center line, and then what do we get?}$$ 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 ...
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2answers
40 views

Group with topology which is not topological group

What will example of a group G with topology such that f: G to G such that f(x) = -x and g: G * G to G such that g((x,y)) = x * y (where * is binary operation on G) both are not continuous.
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1answer
34 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 ...
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1answer
12 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 ...
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1answer
48 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, ...
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2answers
25 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 ...
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22 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 ...
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35 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 $G$?
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2answers
62 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?
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2answers
56 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 ...
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1answer
33 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? ...
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2answers
20 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 $I^{\omega}$ ...
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1answer
49 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?
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3answers
255 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)?
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1answer
31 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 $f(A)$ ...
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1answer
23 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 $S^3$) ...
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0answers
24 views

Complete subspace of continuous function from compact subset [on hold]

Assume $K\in \mathbb{R}$ compact. How to prove that $C^0(K,\mathbb{R})$ is complete. Where $C^0(\mathbb{R},\mathbb{R})$ is the space of continuous f from $\mathbb{R}$.
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1answer
31 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 ...
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0answers
64 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 ...
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4answers
56 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 → ...
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0answers
14 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 ...
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1answer
26 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?
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1answer
36 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 ...
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1answer
49 views

Torus/Moebius Band homeomorphism

Is a fattened Moebius 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 ...
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3answers
61 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
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1answer
32 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 ...
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2answers
38 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. ...
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3answers
38 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 ...
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3answers
33 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 ...