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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85 views

uncountable product of two-point discrete space

I know that uncountably product of $\{0,2\}$ is not-metrizable, separable, compact, Hausdorff and not second countable. But what we can say about Lindelöf property? I think it is not Lindelöf, but how ...
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189 views

Higher homotopy groups!

How would you show that $\pi_n, n>1$ of the Klein bottle is the trivial group? I was thinking Seifert-Van Kampen could be applicable?
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309 views

Show a subset of $\mathbb{R}^2$ is connected.

Question: Show that the set $C=\{(x,y)\in \mathbb{R}^2: 1\leq x^2+y^2<2\}$ is connected. My Question: My main question is what open sets we should pick in $\mathbb{R}^2$. Once I know what open ...
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1answer
244 views

Connected normal space can just be a single point?

Using Urysohn's Lemma, it can be shown that a connected normal space $X$ (with more than one point) is uncountable. But then how can it be that a connected normal space might just be a single point? ...
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498 views

Compact metrizable space has a countable basis (Munkres Topology)

Let X be a compact metrizable space. Would you help me to prove that X has a countable basis. Thanks.
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83 views

$A = \{(x,y) \in \mathbb{R^2}: y = \frac{1}{x}, x > 0\}$. Show $A$ is closed in $\mathbb{R^2}$

Given $A = \{(x,y) \in \mathbb{R^2}: y = \frac{1}{x}, x > 0\}$. Show (by considering convergent sequences or otherwise) $A$ is closed in $\mathbb{R^2}$. Anyone able to give me some advice on how ...
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604 views

Is the union of two nowhere dense sets nowhere dense?

Is the union of two nowhere dense sets nowhere dense? Using the following definition: A nowhere dense set is a subset $E\subset X$ of a metric space (or topological space) $X$ such that ...
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95 views

what is the closure of $\mathbb{Q}^\mathbb{N}$ in $l^\infty$?

I was wondering that since $l^\infty$ is not separable, which means that there is not a countable dense set in it. However the set $\mathbb{Q}^\mathbb{N}$ is countable (am I right in this?). So what ...
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88 views

Every point in a metric space has at least a neighborhood?

I was reading about topology and I came across this statement: Every point $x$ in metric space $(X,d)$ has a neighborhood, which a neighborhood of $x$ (denoted $N(x)$) is defined as there exists ...
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349 views

Subsets of $\mathbb{R^2}$ using the discrete metric are closed, why?

I understand why all subsets of say, $\mathbb{R^2}$ are open with respect to the discrete metric - Let $U$ be a subset of $\mathbb{R^2}$. For all $x \in U$ we can choose an 0 > r > 1 such that we will ...
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149 views

Alternative definition of density

Let $E$ be called dense in $\mathbb{R}$ if and only if $\text{int}(\mathbb{R} \setminus E)=\emptyset.$ Let $x \in\text{int}(\mathbb{R} \setminus E)=\emptyset$. Then for $\epsilon >0$, ...
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105 views

Metric space, subsets and true statements

Let $(X, d)$ be a metric space and let $A$ and $B$ be subsets of $X$. Define $d(A,B) = \inf\{d(a, b) : a \in A, b\in B\}$. Pick out the true statements. a. If $A$ and $B$ are disjoint, then $d(A,B) ...
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306 views

Does there exist continuous map from $S^1$ to $\mathbb{R}$ such that $f(x)=f(y)$ for uncountably many $x,y$?

Does there exist continuous map from $S^1$ to $\mathbb{R}$ such that $f(x)=f(y)$ for uncountably many $x,y\in S^1$? By the Borsuk-Ulam theorem, I know there is no injective map from $S^1\rightarrow ...
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1answer
81 views

What does it mean for a topological space $X$ to have a binary open cover?

Does someone know what's meant by a binary open cover of a topological space $X$? I can't find this definition of binary open cover. Could someone who knows it tell me? Thanks ahead for any help:)
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159 views

On the space of ultrafilters on $N$

I meet the space $X$ of ultrafilters on $N$ with the topology generated by sets of the form $\{p\}\cup A$ where $A\in p \in X$. I can't understand the definition of the topology. Is the points in $N$ ...
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152 views

Why is special about the reverse inclusion in the following example ?

Given a point $x$ in a topological space, let $N_x$ denote the set of all neighbourhoods containing $x$. Then $N_x$ is a directed set, where the direction is given by reverse inclusion, so that $S ≥ ...
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121 views

Is $M=\{(x,y)\in (0,\infty )\times\mathbb{R} : y=\sin(\frac{1}{x}) \}$ a closed set in space $((0,\infty )\times\mathbb{R} ,\rho_{e})$?

Is $M=\left \{(x,y)\in (0,\infty )\times\mathbb{R} : y=\sin(\frac{1}{x}) \right \}$ a closed set in space $((0,\infty )\times\mathbb{R} ,\rho_{e})$, $\rho_{e}$ - Euclidean metric ? I think that open ...
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308 views

set theory question from munkres topology

Munkres topology book set theory chapter 1 question 10c Let $\mathbb R$ denote the set of real numbers. for each of the following subsets of $\mathbb R\times\mathbb R$ determine whether it is equal ...
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809 views

Cantor intersection theorem

I have Cantor intersection theorem: Let $X$ be a complete metric space, and let $\{F_n\}$ be a decreasing sequence of non-empty closed subsets of X such that $d(F_n)$ converges to $0$. Then ...
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1answer
160 views

Cartesian Product of the Real line with a discrete sets

Suppose $S$ is a set of n points, that is $|S| =n$ seen as a discrete smooth manifold. Then is the cartesian product of manifolds $\mathbb{R} \times S \simeq \mathbb{R}^n$? If not what is it?
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132 views

How to interpret “open” without an obvious topology [duplicate]

Possible Duplicate: Why do the $n \times n$ non-singular matrices form an “open” set? I have a topological group (general linear group) $G = \{$ invertible $n\times n$ ...
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122 views

compactness of sets in metric spaces

I couldn't solve this problem. Any help please? Let $X$ be a metric space and $E_{i}\subset X$ for $i=1,2,...$ be nonempty compact sets such that $E_{1}\supset E_{2}\supset E_{3}\supset ...$ Prove ...
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84 views

Covering map of what order?

How can you see in this picture of what order the covering maps be? Well I look for one with a degree of two and one with a degree of four. thanks a lot!
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166 views

A non-denumerable set has non-denumerably many cluster points?

I can´t prove this fact in $\mathbb{R}$. I want to know how general this result is. (What topological properties are needed to prove it?) Let $X$ be a non-denumerable subset of the real numbers. How ...
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1answer
1k views

Arbitrary intersection of closed sets

It can be proved that arbitrary union of open sets is open. Suppose $v$ is a family of open sets. $\cup_{G \in v}G = A$ is an open set. Now by De Morgan's theorem: $(\cup_{G \in v}G)^{c} = \cap_{G ...
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40 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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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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50 views

concept of local base at a point of a topological vector space

I don't understand the concept of local base at a point of a topological vector space. What is the meaning the requirement that any neighbourhood of a point contains some collection from the local ...
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32 views

Fibre is open in covering space

I think I don't see the wood for the trees: In my notes I found the remark that if $p:E \rightarrow B$ is a covering map, then for each $b \in B$ we have that $p^{-1}(b)$ in $E$ has the discrete ...
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42 views

reason of defining continuous function between two topological spaces?

What is the reason of defining continuous function between two topological spaces ? (is it that under continuous function image of a compact/connected set is compact/connected)
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2answers
42 views

Why is the product of homeomorphisms a homeomorphism?

Let $f:A\to B$ and $g:A\to C$ be homeomorphisms, where $A,B,C$ are topological spaces. My book says that $(f\times g):A\to B\times C$ is also a homeomorphism. I wonder why that is. Define ...
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33 views

Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
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38 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
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69 views

Construction of an embedding of $\mathbb{Z} \cup \{\infty\}$ into $\mathbb{R}$.

Let $X$ be the one-point compactification of the integers $\mathbb{Z}$, construct an embedding of $X$ into the reals $\mathbb{R}$. I already appreciate your hints/answers. Thanks
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34 views

an example of a continuous bijection which is not a homeomorphism [duplicate]

I need an example of a continuous bijection $f:X \to Y$, where $X$ is NOT compact and $Y$ is Hausdorff, such that $f$ is not a homeomorphism. (It is easy to show that if $X$ is compact, then $f$ is ...
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74 views

${\overline{A}}^{\circ}= \varnothing \Longrightarrow {\overline{A \cap B}}^{\circ}= \varnothing $

If $X$ is a topological space and $A,B,C \subseteq X$ with $B \subseteq A$, I am wondering if the following statements are true. $(1)$ ${\overline{A}}^{\circ}= \varnothing \Longrightarrow ...
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33 views

Existence of a boundary point

I am not particularly well-versed in topology, so I wanted to check with you whether there exists a much simpler argument to prove the following statement or whether there are problems with my proof. ...
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84 views

What are the elements of a quotient space?

What are the elements of a quotient space? Every definition I see effectively defines the elements of a quotient space as sets. Thus, the topology on the quotient space must be defined in terms of ...
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51 views

How to show $\tau=\{(-\infty,b)|b\in \Bbb R\}\cup\{R,\emptyset\}$ is topology and second countable?

$$\tau=\{(-\infty,b)|b\in \Bbb R\}\cup\{R,\emptyset\}$$ a) Show $\tau$ is a topology on X b) Find point of interior,closure and boundary points of $(-\frac{1}3,0)$ c) Show that ...
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45 views

Which of the following sets are dense in $C[0,1]$

Which of the following sets are dense in $C[0,1]$ with respect to sup-norm topology? $1$. {$f$$\in$ $C[0,1]$ : $f$ is a polynomial } $2$. {$f$$\in$ $C[0,1]$ :$f(0)$=$0$} $3$. {$f$$\in$ $C[0,1]$ ...
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1answer
34 views

Connectedness and Discrete Topology

I know that if a topological space $X$ is connected, then the only subsets of $X$ that are both open and closed are $\varnothing$ and $X$. However, if I take for example the interval $I = [0,1]$ ...
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82 views

Fundamental group of torus by van Kampens theorem

So I am currently going through some lecture notes where the fundamental group of a torus is calculated by van Kampen's theorem: The torus is decomposed into its characteristic fundamental polygon ...
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68 views

How to Show that $S=\{(x,y)\in R^2:x>y^2\}$ is open

Show that $S=\{(x,y)\in R^2:x>y^2\}$ is open quite simple one. We need to choose $\epsilon$ for open balls: $D((x_0,y_o),\epsilon)\subset S$ ,$\forall x_o,y_o\in S$. we can take $\epsilon$ as the ...
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101 views

Why this is the case ! [Product Spaces]

I have come across this post "here" and this question come to my mind ! If $X=\{0,1\}^\Bbb N$, then $X$ is the set of all infinite sequences of zeros and ones, and the topology on $X$ is generated by ...
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34 views

Showing $U$ open in topological group $G$ $\implies$ $gU$ is open

If $G$ is a topological group, and $U$ is an open set in $G$, then do we have that $gU$ is also open in $G$? I know that since $G$ is a topological group, the mappings $\mu: G^2 \rightarrow G$ s.t. ...
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46 views

Is there any space which every two distinct points of it can be separated by two neighbourhoods which one of them is open and the other is closed?

In Hausdorff spaces for every two distinct points $x$ and $y$ there exists a neighborhood $U$ of $x$ and a neighborhood $V$ of $y$ such that $U$ and $V$ are disjoint ($U \cap V = \emptyset$). I like ...
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67 views

Can someone explain what this theorem is saying?

The theorem in the textbook I'm reading states: If $x=(x_1,x_2,\ldots,x_p)$ is any element in $\mathbb{R^p}$, then $|x_i|\le\|x\|\le\ \sqrt{p}\sup\{|x_1|,|x_2|,\ldots,|x_p|\}$. Can anyone ...
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79 views

Proving that a metric space with 3 points can be embedded isometrically into $\mathbb{R}^2$

My definition of an isometric embedding is that if $(M_2,d_1)$ and $(M_2,d_2)$ are metric spaces, then $G:M_1 \to M_2$ is an isometric embedding if $d_2(G(x),G(y)) = d_1(x,y)$ for all $x,y \in M_1$. ...
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47 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
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1answer
38 views

Quick topology question

I confused myself. It is a seemingly trivial question: If $U,V,B$ are sets in a topological space $X$ and $U \subset B$ is open in $B$ and $U = U \cap V$ is it true that $U \cap V$ is open in $B \cap ...