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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0answers
34 views

Is the map that builds the map into the pullback continous with the compact-open topology?

I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map $$Top(T,P) \rightarrow Top ...
1
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0answers
66 views

What does $R^I$ stand for?

In section 30 of Munkres, one exercise states that "Give $R^I$ the uniform metric, where $I=[0,1]$". I guess it's not about powers or something, it's some conventional notation because I've never ...
0
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1answer
65 views

Metrizable Space and continuous extension function

Let A be a closed subspace of a metrizable space $X$, then for every continuous function $f:A\rightarrow I$ and every metric $\rho$ on the space $X$ , I want to show that the formula $$F(x) = ...
2
votes
2answers
121 views

wedge product of projective planes

if we have the wedge product of the real projective plane $P^2$ V $P^2$ Then how would i use Seifert Van Kampens theorem to compute the fundamental group $\pi_1$($P^2$ V $P^2$ ) ? i'm some what ...
1
vote
0answers
92 views

Lindelöf Property and Compact space

Let $X$ be a compact space and $L$ is the smallest family of subspaces of$\,X\,$that contains all closed sets and is closed with respect to countable union and intersection. The question is :- Is ...
1
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2answers
119 views

How to prove that, the set of all matrices $M_n{\mathbb R}$ with distinct eigen values is dense in $\mathbb R^n$?

How to prove that the set of all matrices $M_n{\mathbb R}$ with distinct eigen values is dense in $\mathbb R^n$? Is there any geometric interpretations behind this.. if it is so, then tell me how to ...
0
votes
1answer
33 views

any complete metric space $S$ can be homeomorphically embedded as a dense subset of a compact metric space $\bar{S}$

How to prove that any complete metric space $S$ can be homeomorphically embedded as a dense subset of a compact metric space $\bar{S}$. I know that a polish space is homeomorphic to a $G_{\delta}$ ...
1
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0answers
71 views

Levels sets of a continuous function

Suppose $f:[0,1]\rightarrow [0,1]$ is continuous. Let $A$ be the set of all maximal, connected subsets of the level set $f^{-1}(0)$. Can $A$ be uncountable?
1
vote
1answer
875 views

Need example for a topological space that isn't (T1,T2,T3), but is (T4) [closed]

How can gives me an example for a topological space that : $(T_4)$ but it isn't $(T_3)$ , $(T_4)$ but it isn't $(T_2)$ , $(T_4)$ but it isn't $(T_1)$
1
vote
1answer
52 views

Answer gap-filling-in topology, describing the kernel from the Seifert–van Kampen theorem

The question is: Let $X=S^1\times I$ and let $A=S^1\times[0,3/4)$ and $B=S^1\times(1/4,1]$ So that $\{A,B\}$ is an open cover. I have been tasked with using the the Seifert-van Kampen theorem to ...
1
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1answer
22 views

I want to prove that a topological space is Lindelöf

Well, I'm trying to prove the following exercise: Let $X$ be a compact Hausdorff space and its cardinality is greater than the real numbers. Then $X$ is Lindelöf but not separable. Could someone try ...
1
vote
1answer
48 views

Let $\tau=\{(-\infty,a)\ :\ a\in\mathbb{R}\}\cup\{\emptyset, \mathbb{R}\}.$Does $\{\frac{1}{n}\}_n$ converge or diverge in this topology.

Let $\tau=\{(-\infty,a)\ :\ a\in\mathbb{R}\}\cup\{\emptyset, \mathbb{R}\}.$ Discuss the convergence or divergence of $\{\frac{1}{n}\}_n$ in this topology. If $\{\frac{1}{n}\}_n$ converges to $p$, ...
1
vote
1answer
70 views

Component-wise connected sum of links

Given two links $K = K_1 \cup \dotsb \cup K_n$ and $L = L_1 \cup \dotsb \cup L_n$, where each $K_i$ and $L_j$ are oriented knots, can we define the connected sum $K\#L$ by taking the connected sum of ...
0
votes
0answers
76 views

Sequential compactness in weak topology

When the Banach space $V^*$ is reflexive, we have the unit ball in $V^*$ is weak$^*$ sequentially compact. For a Banach space $V^*$ that might not be reflexive, we have to assume that $V$ ...
0
votes
1answer
74 views

Compactness and cartesian product

I'm having trouble figuring out how can I show that if two sets are compact then their cartesian product is also compact. Any help is much appreciated,thank you!
10
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1answer
214 views

Tensor products in general topology?

Let $(X,\tau)$ and $(Y,\sigma)$ be topological spaces and let $(X\times Y,\tau\times\sigma)$ be the space with the box topology. Since I never heard of it I guess that there is no space $X\otimes Y$ ...
1
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2answers
45 views

Subbases of a topology

Let $\mathscr { S } $ the class of all closed intervals $[a,b]$, where $a$ and $b$ are rational , i.e. $a,b\in \mathbb{Q}$ and $a<b$ . Show that $\mathscr{S}\cup \{\{p\}:p\in\mathbb{ Q }\}$ ...
0
votes
3answers
104 views

homeomorphism between $[0,1]\times (0,1)$ and $[0,1)\times(0,1)$

Can someone give a proof that why two spaces $[0,1]\times (0,1)$ and $[0,1)\times(0,1)$ are not homeomorphic?
0
votes
1answer
30 views

Continuous Maps: Totally Boundedness

Obviously, the image of a separable space under a continuous map is separable again: $$\overline{A_0}=X:\quad f(X)=f(\overline{A_0})\subseteq\overline{f(A_0)}\subseteq ...
4
votes
1answer
145 views

Property of constant function

I learn real analysis and topology then I found something interesting about constant function. I am unsure it is true or false because I cannot prove it. I found property as follows: If $X$ is $T_1$ ...
0
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2answers
30 views

Bases local in topology

Prove that if a point p has a finite local base Bp, then it also has a local base consisting of only a set. Can you give me any suggestions?
0
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1answer
38 views

The set of all sections of a frame bundle

Let $\pi:E\to M$ be a smooth vector bundle of rank $n$ and $\pi_{F} : FE \to M$ be associated frame bundle to $E$. The set of all sections of a frame bundle is not a vector space, why? Why is it a ...
0
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2answers
273 views

Intersection of infinitely many closed nested sets in a complete space

$(X,d)$ is a complete metric space. It needs to be proved, that if $A_1\supset A_2\supset ...$ are non-empty closed sets and $A_n$ is a finite sum of closed sets with diameter $\leq 1/n$ then ...
3
votes
3answers
265 views

Need example for a topological space that isn't connected, but is compact

We know the topological space $(R,τ_1)$ is a connected space but it is not compact, $(R,τ_+)$ (which generated by $[a,b[$) is not connected space and it is not compact space, and $(R,τ_{cf})$ is ...
1
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0answers
62 views

How is this topological space different from the euclidean one?

I'm preparing for my topology exam and came across this example which I can't figure out. Let $\mathcal{T}$ be a such family of all sets $U\subset \mathbb{R}^2$ that $U\cap L$ is an open set in L, ...
1
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1answer
147 views

The Kuratowski Monoid

I have been reading the paper "The Kuratowski closure complement theorem" by "B. J. Gardener and M. Jackson". In that the author discusses the 6 different monoid structures as follows: Extremally ...
0
votes
2answers
137 views

an example of the Scott topology

Let $X=A\cup B$ where $A$ is set of all positive integers w.r.t the Scott topology and $B$ is singleton set $\{b\}$. A set $U$ is open in $X$ iff $U$ is open in $A$ or $A\subseteq U$ and $U\cap B ...
3
votes
1answer
68 views

$\{1/n\}_n$ converges in the Sorgenfrey line

The following is my attempted proof that the sequence $\{ \frac{1}{n} \}_n$ converges in the Sorgenfrey line. Offer criticism, please! Consider ...
2
votes
1answer
87 views

$\{1-\frac{1}{n} \}_n$ does not converge in the Sorgenfry line

I am trying to prove that the sequence $\{1-\frac{1}{n} \}_n$ does not converge in the Sorgenfry line. Below is my attempt. Consider ...
1
vote
2answers
117 views

Continuous Extension Mapping

Let assume $A$ is a dense subspace of a topological space $X$ and $f$ is a continuous mapping of $A$ to a regular space $Y$. the question is :- Show that the mapping $f$ has a continuous extension ...
0
votes
1answer
83 views

Compactness of, and the existence of certain subspaces of, the Helly space [closed]

Taking $I = [0,1]$, the Helly space is the subspace of $I^I$ (with the usual Tychonoff product topology) consisting of all nondecreasing functions. I have three questions about this space. Is the ...
1
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2answers
62 views

An example of a not locally compact space in $\mathbb R^2$

Are the two subspaces $X$ and $\operatorname{cl}(X)$ of Euclidean space $\mathbb R^2$ locally compact? $$X = \{(x,\sin 1/x) \mid 0 < x \le 4\}\cup\{(x,\sin 1/x) \mid -4 \le x \lt 0\} \cup ...
2
votes
0answers
63 views

Limit topology of a sequence of topological vector spaces

Under which circumstances is the limit topology of an increasing sequence $E_0\subseteq E_1\subseteq E_2\subseteq\cdots$ of topological vector spaces, where the inclusion maps are linear and ...
1
vote
1answer
91 views

A problem in locally compact Hausdorff space

I am trying to solve the following problem. Let $X$ be locally compact Hausdorff and $Y$ be Hausdorff. (a) If $f: X \to Y$ is continuous and open map then show that $f(X)$ is locally compact. (b) ...
3
votes
1answer
59 views

are there different notions of 'boundary' in the manifold sense and the topological space sense?

this seemingly innocent question has been bugging me for quite a while. Lets give a minimal example: The unit circle S has no boundary considered as a manifold (all points have neighborhoods ...
2
votes
2answers
216 views

one point compactification of discrete space

Problem: What is the one point compactification $X^*$ of a discrete space $X$. In the case of $X$ being finite, $X$ itself is compact so the one point compactification would be merely $X$ $\bigcup$ ...
0
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2answers
49 views

Cardinality of a discrete set, in a separable space.

Given a separable space $X$, if $A$ is discrete subspace of $X$, then $|A|\leq 2^{\aleph_0}$. Some ideas?. It's similar to "jone's lemma", but without $A$ being closed. Whit what addiotional ...
1
vote
1answer
58 views

some statements on sum of two subsets of plane. open, closed etc .

$W=\{(x,y)\in\mathbb{R}^2: x>0,y>0\}$ $X=\{(x,y)\in\mathbb{R}^2: x\in\mathbb{R},y=0\}$ $Y=\{(x,y)\in\mathbb{R}^2: xy=1\}$ $Z=\{(x,y)\in\mathbb{R}^2: |x|\le 1,|y|\le 1\}$ $W+X$ is open ...
5
votes
3answers
170 views

Totally disconnected topologies on countable set.

Are there totally disconnected topologies $\tau$ on a countable set $X$ such that $(X,\tau)$ is not homeomorphic to one of the following? $\mathbb{N}$ with the discrete topology; one-point ...
2
votes
2answers
53 views

a question about connected set, how to know whether A is connected or not?

In the Euclidean plane $R^2$,consider the subset $$ A=\{(x,y)\in \Bbb R^2|\text{Either $x$ or $y$, but not both, is a rational number}\} $$ Is $A$ connected? Is $\Bbb R^2$\A connected? I have ...
0
votes
1answer
69 views

Is the definition of continuity in analysis a particular case of topological continuity?

Take a constant function and remove an open interval from it: $$f(x)= 1, \text{if $x\in(-\infty,0]\cup[1,\infty)$ }$$ This function shouldn't be continuous because at $0$ no right limit of the ...
4
votes
2answers
148 views

Show that the set of all complex numbers $z$ such that $|z| \leq 1$ is closed?

I'm working through Rudin's "Principles of Mathematical Analysis" on my own, so I don't want the full answer. I'm only looking for a hint on this problem. Rudin states without proof that the set $X = ...
3
votes
0answers
49 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
3
votes
0answers
103 views

A question about compact sets: how to prove $g$ must be an isometry [duplicate]

Let $(X,p)$ be a compact metric space. Suppose that $g:X\rightarrow X$ is a function such that for all $x_1,x_2\in X$ we have $p(g(x_1),g(x_2))\geq p(x_1,x_2)$. Prove that, in fact, $g$ must be an ...
1
vote
2answers
82 views

Are $[0,1)\times [0,1)$ and $[0,1]\times [0,1)$ homeomorphic?

Are $[0,1)\times [0,1)$ and $[0,1]\times [0,1)$ homeomorphic? Not getting any idea how to start.
0
votes
1answer
54 views

What is the largest complete subspace of $(\mathbb{Q}, |\cdot|)$

For example $\left\{\frac{1}{n}\right\}\cup \{0\}$ is a complete subspace of $\mathbb{Q}$, but I am having trouble writing out the largest (in the sense of "$\subset$") complete subspace in ...
1
vote
1answer
56 views

a question about compact set, how to prove there exits f(y)=y [duplicate]

Let (X,p) be a compact metric space.Suppose that f X->X is a function such that, for all $x_1$,$x_2$ $\in$X, if $x_1\neq x_2$ then p(f($x_1$),f($x_2$))<$p($x1$,$x2$)$. Prove that there exits a ...
3
votes
2answers
57 views

One point compactification of $\Bbb{R}\setminus \{0\}$

What will be one point compactification of $\Bbb{R}\setminus \{0\}$? It looks like it will be union of two circles touching at a point. But do I write a Mathematical proof to justify my claim?
0
votes
2answers
26 views

$x$ x $1/x$ for $\epsilon$ $\gt 0$ has no $\epsilon$-neighborhood in $R_{+}$ x $R_{+}$

This is a problem from Munkres' Topology. Define the $\epsilon$-neighborhood of $A$ in a metric space $X$ to be the set $U(A, \epsilon) = ${$x$ | $d(x,A)$ $\lt$ $\epsilon$}. (d) Assume that $A$ is ...
1
vote
1answer
39 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim \operatorname{diam} E_n ...