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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2answers
31 views

Prove the existence of minimal height of a convex polygon

Suppose we have a polygon in $\mathbb{R}^n$. Obviously, we can always trap the polygon into two parallel hyperplanes perpendicular to any given direction. Now the question is, how to prove the ...
1
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0answers
40 views

closedness of compact sets in some topological spaces

Is there any famous axiom on X other than Hausdorffness or axioms leading to Hausdorffness,such that every compact set in X is closed?
7
votes
1answer
40 views

When does a topological space inherit multiplication from a dense subspace?

Suppose $K$ is a compact topological Hausdorff space with a dense subspace $G$. Moreover, let $G$ have a group structure which is compatible with the topology inherited from $K$. i.e. $G$ is a ...
0
votes
1answer
24 views

Why can we consider elements of a normed space $X$ as elements of a normed space $Y$, if there is an embedding between these spaces?

Let $(X,\left|\;\cdot\;\right|)$ and $(Y,\left\|\;\cdot\;\right\|)$ be normed spaces and $\iota :X\hookrightarrow Y$ be an embedding. Often when I read that such an embedding $\iota$ exists, I read ...
2
votes
1answer
41 views

Metric spaces as Cauchy complete categories, nlab entry, insight into a few of the constructions.

I'm having a bit of trouble making sense of some of the concepts in the "Metric space" section on nlab's entry on "Cauchy complete category" ...
1
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2answers
30 views

Question about functions and topology

This is a very general question, but one that I have been struggling with. If we say that a function from a topological space X to a topological space Y is ONTO, then does that mean that for each open ...
1
vote
1answer
23 views

Find the number of relatively compact connected components

Let $X=\{(x,y):x^2+y^2<5\}$ and $K=\{(x,y):1\leq x^2+y^2\leq 2 or 3\leq x^2+y^2\leq 4\}$. Then: Find the number of relatively compact connected components of $X\setminus K$ in $X$. I drew a ...
1
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0answers
34 views

Question about limit points

The reference I'm using defined a limit point of a set as: Definition. A point $x$ is a limit point of a set $A$ if every $\epsilon$-neighborhood $V_{\epsilon}(x)$ of $x$ intersects the set $A$ in ...
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0answers
35 views

Existence of an homeomorphism between $X$ a complete separable metric space and a subspace of $[0,1]^{\mathbb{N}}$

Result: If $X$ is a complete separable metric space then there is a $E \subset [0,1]^{\mathbb{N}}$ such that $X$ is homeomorphic to $E$ ($E$ is a $G_\delta$ set - is the intersection of denumerable ...
2
votes
0answers
53 views

True/ false questions of basic concepts in topology

Hi the following are only true or false questions and I'd appreciate if someone can help me to check if my answers are correct. Thanks :) (a) Any set of $X$ admits a metric such that the induced ...
3
votes
1answer
53 views

What do open sets look like in this topological space?

I recently encountered the following: Consider the following space: the underlying set is $C = C_1 \cup C_2$, where Ci is the circle of radius $i$ and centre $0$ in the complex plane. Basic open sets ...
2
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0answers
12 views

Sheafification and restriction to open subset

Let $X$ be a topological space and $\mathcal{F}$ be a presheaf on $X$. We denote by $\mathcal{F}^+$ the sheafification of $\mathcal{F}$. Let $U\subset X$ be an open subset. We denote by ...
-1
votes
3answers
39 views

I am confused, is $\{[a,b], \phi\}$ a topological space based on the definition of the open set?

From the definition based on neighbourhood, I know the interval $[a,b]$ is not belong to a topological space, as the end points have no neighbourhoods; However, I can not judge it based on the ...
1
vote
1answer
45 views

If $M$ is Borel and $M=M_1\times M_2\times\cdots \times M_n$, then $M_i$'s are Borel?

If $M$ is a nonempty Borel set in $\mathbb{R}^n$ and $M=M_1\times M_2\times\cdots \times M_n$, then are $M_1,M_2,\ldots,M_n$ are Borel sets in $\mathbb{R}$? I think the answer is yes. Using ...
-2
votes
3answers
66 views

Poincare conjecture 'loop' contraction issue

According to WikiPedia's entry on Poincare's conjecture, the first image (on the right - five spheres): For compact 2-dimensional surfaces without boundary, if every loop can be continuously ...
1
vote
1answer
25 views

Are Normed linear spaces $T_4$(Normal Hausdorff)?

Do normed linear spaces have the properties of a normal hausdorff space? I just sat an exam and I couldn't work out how to prove something initially, then I assumed that normed linear spaces are ...
1
vote
1answer
29 views

The topology on $X / G$ where $G$ acts on $X$

The elements are orbits, but how do we find the neighbourhoods? In particular, let $G= ( \mathbb R^+ , \cdot )$ and $X=[0, \infty )$. Let $G$ act on $X$ via the usual multiplication. Then $X/G = ...
15
votes
2answers
279 views

Continuous surjections onto $\mathbb{R}$

I have two questions about continuous functions: Suppose $X \subseteq \mathbb{R}$ and $X$ has same cardinality as $\mathbb{R}$. Can we find a continuous function from $X$ onto $\mathbb{R}$? Suppose ...
3
votes
1answer
61 views

Examples and counter-examples in Real analysis - check my answers please

I've been given some practice examples, without solutions in preparation for an upcoming exam, and was hoping I could get them double checked here. For each of the following, either give an example ...
7
votes
1answer
73 views

Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: Why we need continuity to show the result?

Let $f: X\mapsto Y$ be a closed continuous surjective map such that $f^{-1}(y)$ is compact, for each $y\in Y$. Show that if $Y$ is compact, then $X$ is compact. My question is why do we need $f$ to ...
0
votes
2answers
36 views

Difference between topology and sigma-algebra axioms.

One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under ...
5
votes
1answer
85 views

Curve meeting itself everywhere

(related, but not a duplicate: curve which crosses itself at every point ) When reading the comments to the question above, it has been pointed out that if by "cross" we mean that for every ...
3
votes
1answer
46 views

Is a compact subset of a topological group G/N closed if G is hausdorff?

I just used this hypothesis when I was proving a theorem.But I was not sure if this hypothesis is correct.
0
votes
1answer
28 views

Prove that a function is continuous using basic open sets

Using basic open sets of $\Bbb R$, prove that $f(x,y,z)=x^2+y^2+z^2+2x+2y+6$ is a continuous function from $\Bbb R^3$ to $\Bbb R$. My attempt: Since $f(x,y,z)$ is continuous and $f(x,y,z)\in ...
2
votes
1answer
77 views

Prove: $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components.

I need to prove or at least to understand why $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components. But I've no idea how to deal with it. I even tried to draw it ...
3
votes
1answer
49 views

Connectedness of the Hausdorff distance.

Does anyone know a proof of connectedness of the Hausdorff distance? I mean a proof of the following: Theorem If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected. ...
3
votes
2answers
33 views

What is “approximate compactness”? What is an example of an approximately compact set?

I read this: A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a ...
4
votes
2answers
27 views

sequence of open sets

Find the sequence of open sets in $\Bbb{R}$ like $\{G_n\}$ such that $\Bbb{Z}=\bigcap _{n=1} ^{\infty}G_n$. I think an answer is this: $$G_n=\bigcup_{m=1} ...
4
votes
0answers
39 views

Convergence and integral for nets.

We say that the non-empty set $S$ with partial order $\leq$ is directed set if for any $s,t\in S$ we have a $u\in S$ such that $s\leq u, t\leq u$. A net is a function from directed $S$ into the any ...
1
vote
1answer
49 views

An simple example to show that every countably compact space needn't be compact

I am willing to study compact and connected in topological space and apply in other topological spaces. I am a beginner in this subject. Kindly give some examples. I have went through few books but I ...
0
votes
2answers
31 views

Compact Hausdorff spaces are normal

I want to show that compact Hausdroff spaces are normal. To be honest, I have just learned the definition of normal, and it is a past exam question, so I want to learn how to prove this: I believe ...
1
vote
1answer
20 views

Proof attempt for collection of all open intervals being a basis of $\Bbb R$ with the standard topology

Show that the collection of all open intervals $\{(a,b)\}$ is a basis of $\Bbb R$ with the standard topology: My attempt: I believe we want to show two things: 1) All elements, $x\in\Bbb R$ are ...
1
vote
2answers
36 views

Proving that a compact subset of a Hausdorff space is closed

I am having trouble understanding the answers here. I am trying to prove that a compact subset of a Hausdorff space is closed. Following the proof is difficult, perhaps because Brian reused letters ...
0
votes
1answer
22 views

A uniform space must be a symmetric space.

The topology of a uniformizable space is always a symmetric topology; that is, the space is an $R_{0}$-space. How to prove it? It must be a simple question, but I can't write it down. $X$ is an ...
4
votes
1answer
32 views

Does this implies that two metric spaces are Equivalent?

If two metrics $d_i$ on the same set $X$ have the same Cauchy sequences (ie. if a sequence is Cauchy for the first metric, it is also Cauchy for the other one and vice versa).Does this imply that the ...
1
vote
2answers
40 views

Show a set is a closed set larger than open set

let $(\mathbb{R}^n, \tau)$ where $\tau$ is the std topology. Show that the set $U = D_{\delta}(x) = \{y\in \mathbb{R}^n: ||y-x||\leq \delta \}$ is a closed set. so under these conditions it means I ...
2
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0answers
41 views

Deck transformation

I read that a deck transformation is uniquely defined by the value of one point. Unfortunately, I don't understand where this comes from. I mean, all we know is that there is one point in the fibre ...
0
votes
2answers
42 views

A point is in $\partial A$ iff every neighborhood of it contains both a point of $A$ and a point of $X\setminus A$.

How to prove (c) A point is in $\partial A$ iff every neighborhood of it contains both a point of $A$ and a point of $X\setminus A$. (d) A point is in $\operatorname{cl}(A)$ if and only if ...
1
vote
1answer
26 views

Show $f(x) = (x,y_0)$ is an embedding where $x \in X, y_0 \in Y$ (fixed) for $X,Y$ topological spaces.

I know definition of embedding. $f: X \to X \times Y$ $f(x) = (x,y_0)$ is a horizontal strip in $X \times Y$ I need to show $f,\pi_1$ where $\pi_1: X \times Y \to X$ by $\pi_1(x,y) = x$ are both ...
-3
votes
0answers
27 views

countable open cover is reducible to a finite cover [duplicate]

Let X be a T1-space. prove that X is countably compact if and only if every countable open cover of X is reducible to a finite cover.
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2answers
50 views

Proposed proof of topological result

Hi I just want to find out if the following is a acceptable proof for the proposition: "Consider metric space $(X,d)$. If $A \subset X$ is connected and $A \subset B \subset \bar{A}$ then $B$ is ...
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votes
0answers
36 views

Distance between a point and a nonempty subset of Euclidean space

let $E \subset \mathbb{R}^n$ be a non empty set. The distance between the point $a$ and the set $E$ is defined as $qE(a)=\inf\{||x-a||: x \in E\}$. 1) let $A=\{(q_1,q_2): q_1,q_2 \in Q$ and ...
2
votes
1answer
16 views

Adjunction space of $D^2 \cup_f S^1$ is homeomorphic to $\mathbb{P}^2(\mathbb{R})$

Suppose we have the attaching map $f: S^1 \rightarrow S^1, z \mapsto z^2$. I am trying to show that $D^2 \cup_f S^1$ is homeomorphic to $\mathbb{P}^2(\mathbb{R})$. Could someone give me a tiny hint? ...
3
votes
1answer
50 views

Why is $\Bbb R\setminus\{\frac1n\mid n\in\Bbb N\}$ not locally compact?

I have a question: if I take in $(\mathbb{R},|.|)$ the set $A=\left\{\frac1n, n\in \mathbb{N}\right\}$ and I consider the set $B=\mathbb{R}\setminus A$ I want to prove that $B$ is not locally ...
4
votes
3answers
61 views

A continuous surjection from irrational numbers to Cantor's set

I wonder if there exists such a function that is continuous and surjective $$f:\mathbb{R} \setminus \mathbb{Q} \rightarrow C$$ where $C$ is the Cantor's set. When I did such an exercise but for $f:C ...
2
votes
1answer
28 views

Is $(\mathbb{Q}\times\{0\}) \cup (\mathbb{R}\times(0,\infty))$ $F_{\sigma},G_{\delta}?$

Is A = $(\mathbb{Q}\times\{0\})\cup (\mathbb{R}\times(0,\infty))$ $F_{\sigma},G_{\delta}?$ I started with saying that it is not $G_{\delta}$ because: Let's say that it is $G_{\delta}$. Then ...
1
vote
1answer
25 views

Why is this image sequentially compact? [duplicate]

Assuming $X$ and $Y$ are normed spaces, $K\subset X$ and $f:K\rightarrow Y$. Why is the image $f(K):=\{f(x)\in Y: x\in K\}$ sequentially compact, if $K$ is sequentially compact and $f:K\rightarrow Y$ ...
0
votes
2answers
29 views

Why is this function uniformly continuous?

Assuming $X$ is a normed space. Why is a function $f:K\rightarrow\mathbb{R}$ uniformly continuous on a subspace $K\subset X$, if $K$ is sequentially compact and $f$ is continuous?
1
vote
1answer
61 views

Properties of Pushout

suppose we have a pushout square in $\mathrm{Top}$: \begin{align*} \require{AMScd} \begin{CD} X_0 @>{\mu_1}>> X_1\\ @V{\mu_2}VV @VV{\alpha_1}V \\ X_2 @>>{\alpha_2}> X ...
5
votes
3answers
120 views

Can this be a way to prove that $\Bbb{R}^2$ and $\Bbb{R}^3$ are not homeomorphic?

The normal way I use to prove that $\Bbb{R}$ and $\Bbb{R}^2$ are not homeomorphic is by removing a point and then using path connectedness. But this method doesn't seem to work for $\Bbb{R}^m$ and ...