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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24 views

Topological , Homeomorphic version of $|S \times S|=|S| $

Give example of a subset $A$ of $\mathbb R$ such that with respect to some topology , $ A$ is homeomorphic to $A\times A$ . In set theory ZF it is known to be equivalent to A.C. that for any ...
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0answers
24 views

algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists ...
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2answers
33 views

If two sets are separated, then any two subsets of those sets are also separated?

I want to prove that if two sets X and Y are separated, then subsets of those sets are also separated. The definition is that if X intersect Y closure is empty and X closure intersect Y is empty, the ...
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2answers
29 views

Closed set on topological space [duplicate]

This is a problem on topological spaces and continuous functions. If $f,g \to\mathbb{R}$ are continuous functions, then $T=\{x\in X: f(x)=g(x)\}$ is closed on X
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0answers
19 views

Question about dimension in Notherian spaces

Let $X$ be a Notherian topological space of finte dimension which is Kolmogorov (meaning that for two points $X$ there exists an open subset of $X$ containing one of them but not the other). This ...
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2answers
68 views

Is every metric space subspace of some connected metric space?

If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected .I'm not sure whether this will work or not, but intuitively I think the ...
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2answers
31 views

Open connected subset of $ \mathbb R^2 $is path connected [duplicate]

Is open connected subset of $ \mathbb{R^2} $ is path connected?
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1answer
27 views

A set of real numbers whose limit points from a countable set

Construct a set of real numbers whose limit points from a countable set. Is the set you constructed closed? Is it compact? My example is $$G=\{1/n+1/m: n, m \in \mathbb N\}\cup \{0\}$$ and as ...
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0answers
12 views

When is a metrizable topological vector space locally bounded?

Consider a topological vector space $E$ with topology $\sigma$. Suppose that $E$ is metrizable, in other words, that there exists a metric $d$ on $E$ that induces the topology $\sigma$. One can then ...
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2answers
23 views

Klein bottle contains Mobius band

I read the following: "The Klein bottle contains a copy of the Mobius band". I assume this means that there is a subspace of the Klein bottle that is homeomorphic to the Mobius band. How do we obtain ...
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0answers
10 views

Orientability of Bordered Presentation and its Closure

How can the following claim be true? If $\Pi $ is a bordered presentation, then $\Pi ^c$ is orientable if and only if $\Pi $ is orientable. We know that $\Pi$ must contain border arcs. By ...
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1answer
16 views

Connected sum of projective plane $\cong$ Klein bottle

How can I see that the connected sum $\mathbb{P}^2 \# \mathbb{P}^2$ of the projective plane is homeomorphic to the Klein bottle? I'm not necessarily looking for an explicit homeomorphism, just an ...
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1answer
26 views

Show compactness of a set given by inequalities

Show that the subset $A=\{(x_1,...,x_n)\in\Bbb R^n |−1≤x_1 ≤x_2 ≤···≤x_n ≤1\}$ is compact. A is contain in an open cover as it is contained in $\Bbb R^n$. Therefore there exists a finite sub cover ...
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1answer
19 views

$t$-adic topology (on $\mathbb F_p(1/t)$)

Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic topology. The $t$-adic ...
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0answers
15 views

Performing an Excision on a Topological Surface

Recently I began a book on topology, but the concept of excision on a topological surface isn't clear; perhaps you, collectively, could help elucidate it. Suppose we have an arbitrary topological ...
5
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1answer
45 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
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0answers
19 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
26 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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33 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
40 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
19 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
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1answer
47 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
25 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
27 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
37 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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0answers
12 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
35 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 ...
3
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0answers
20 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
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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 ...
2
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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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2answers
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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
52 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
22 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
33 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
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2answers
58 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
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1answer
49 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 ...
4
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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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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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1answer
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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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0answers
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 $G$?
3
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2answers
67 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
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2answers
61 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
35 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 $I^{\omega}$ ...
3
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0answers
39 views
+500

On infinite groups admitting finitely many group topologies

It has been proved there is an infinite group which admits exactly two group topologies [1]. For which $n$, is there an infinite group $G$ which admits exactly $n$ group topologies ordered linearly ...