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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4
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1answer
67 views

Fundamental group of the complement of an eight shape

What is $\pi_1(\mathbb{R}^3 \setminus (S^1 \vee S^1))$? I would say that the space deformation retracts to a sphere $S^2$ surrounding the missing eight with two sticks stuck in the two loops of the ...
0
votes
2answers
22 views

Why does the second requirement of a Basis for a Topology make sense?

Why does the second requirement of a basis, i.e. that a point in two sets must also be contained in a third set contained in the intersection of the two set, make sense? Why is that what we should ...
0
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0answers
31 views

How to prove a rectangle is compact

Show that the rectangle $K=[0,a]\times [0,b]$ is a compact subset of $\mathbb{R}^2$. My try : Take some open cover $\{U_{\alpha}\}_{\alpha\in \Lambda}$ of $K$. Now I tried to prove it is closed and ...
3
votes
1answer
29 views

“Heine–Borel” for the Sorgenfrey line [duplicate]

The Heine–Borel theorem perfectly characterizes the compact subsets of the real line $\mathbb{R}$ (with the usual metric/order topology): Heine–Borel Theorem. A subset $A \subseteq \mathbb R$ is ...
-1
votes
3answers
27 views

Let $O_1$ and $O_2$ be open sets in $R$, prove that $O_1 \times O_2$ is an open set in $R^2$

How would you prove the following? If $O_1$ and $O_2$ are open sets in $R$, then the set given by $O_1\times O_2$ is an open set in $R^2$ I'm understanding open as: $O$ is open if every point of ...
0
votes
1answer
24 views

Characterizing the family of Borel subsets of a subspace

Given any topological space $X$, let $\mathcal{B}(X)$ denote the $\sigma$-algebra of Borel subsets of $X$. Let $X$ be a topological space and let $Y\subset X$ be given the relative topology. Then ...
1
vote
1answer
36 views

Separable implies second countable

We have $(X,d)$ a metric space. The problem I want to prove is quite long so I'll just put what I need to get it: if $X$ is compact then is separable if $X$ is separable then is second countable ...
0
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2answers
26 views

Showing that we can represent a finite union of intervals as a finite union of pairwise disjoint intervals

Let $K$ be a family of subsets of $\mathbb{R}$ s.t. $A \in K$ iff $A$ can be represented as a finite union of intervals of the form $[x,y)$. Show that $\forall A \in K$ can be represented as a finite ...
1
vote
1answer
67 views

general topology (self learning)

Hi everyone I'd like to know if the following is correct. I'd appreciate any suggestion. Thanks in advance. From Dudley´s book: Let $A_n$ be the set of all the integers greater than $n$. Let ...
0
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0answers
28 views

Does continuity follow from linearity on all or only finite-dimensional vector spaces

I'm currently reading an introduction book on topology. While solving one of its exercises I came across something odd. The exercise is: Let $E$ and $F$ be normed spaces, let $T:E \to F$ be linear, ...
1
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0answers
30 views

Is there a non-empty subset of $\mathbb{R} $ like $A$ such that the set of accumulation points of $A$ is itself and $A\cap\mathbb{Q} = \emptyset $ [duplicate]

Is there a non-empty subset of $\mathbb{R} $ like $A$ such that the set of accumulation points of $A$ is $A$ and $A\cap\mathbb{Q}=\emptyset\,$?
0
votes
1answer
46 views

Is the pullback of two covering spaces $\tilde X$ and $\hat X$ a covering space?

Suppose we have two covering spaces $p:\tilde X \rightarrow X$ and $q:\hat X \rightarrow X$ of the same space. Is the pullback $\tilde X \times_X \hat X$ also a covering space of $X$? If yes, what ...
0
votes
2answers
78 views

Is there a continuous surjection from the closed unit square $[0,1]\times[0,1]$ to $\mathbb R ^2$?

Is there a continuous surjection from the closed unit square $[0,1]\times[0,1]$ to $\mathbb R ^2$? If yes, please give examples. I'm a little stuck on this. What if I replace the closed unit ...
3
votes
0answers
29 views

Compactness in minimax theorem

According to Von Neuman's minimax theorem we have $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$ for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
0
votes
3answers
39 views

A map $f:([a,b], |\cdot|) \to ([c,d], |\cdot|)$ is an isometry if and only if $d-c = b-a$.

I was asked to prove the following problem: A map $f:([a,b], |\cdot|) \to ([c,d], |\cdot|)$ is an isometry if and only if $d-c = b-a$. But I think this is not correct, specifically the sufficient ...
0
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0answers
28 views

Is the number of open sets same in homeomorphic topological spaces?

Does the homeomorphic topological spaces, have same number of open sets?
-2
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0answers
20 views

The closure of $A=\{x\times\frac{1}{2}|0<x<1\}$

I want to find the closure of the following set of ordered square topology: $$A=\{x\times\frac{1}{2}|0<x<1\}$$
0
votes
1answer
35 views

How to prove the equivalence of 2 affine spaces given that one is the subset of the other one?

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
1
vote
0answers
41 views

open sets in the order topological space

I have a question. I am really confused about determining if a set is open. First, the idea of a set being closed has nothing to do with homomorphic ideas of closure: if $x,y \in F$ then $x+ y$ ...
1
vote
0answers
29 views

Introductory Topology True/False check - Topology without tears - Exercises 1.1

I have just started to learn Topology, using specifically the book mentioned in the title. I have placed that information in the title with SEO in mind, if this is not acceptable practice in this ...
0
votes
1answer
40 views

strictly finer topologies and bases

i have just a quick question, if T1 > T2 where T1 is strictly greater than T2 are their respective bases strictly greater? where T_i are topologies on a set X . i dont think so because since ...
3
votes
1answer
84 views

Questions about simplex and affine space

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
0
votes
0answers
33 views

The closure of $\{x\times0\mid 0<x<1\}$

I just want to know if I am right The closure of $\{x\times0\mid 0<x<1\}$ is $\{[0,1]\times0\}$ with the ordered square topology, is that right?
0
votes
1answer
35 views

triangle inequality for a metric space

If $d_{\infty}(a,b) =$ max$\{|a_{i} - b_{i}|\}$ for $1 \leq i \leq k$, I want to prove that this is a metric on $\mathbb{R}^k$. Its pretty clear that $d_{\infty}(a,a) = 0$ and it is also pretty clear ...
4
votes
2answers
57 views

Prove that the set $\{(x,y) \in\mathbb R^2\mid y<x^2\}$ is an open set by giving an explicit radius

I´m basically trying to prove that if $S$=$\{(x,y) \in\mathbb R^2\mid y<x^2\}$. Then, for any $(x,y) \in S$ there exists a radius $\delta$ such that $B_\delta(x,y) \subseteq S$. What values of ...
1
vote
1answer
52 views

$S^{n-1}$ is not a deformation retract of $\mathbb{P}^n(\mathbb{R})/ B(0,1)$.

Let $n$ be $\geq 2$, $$\mathbb{P}^n(\mathbb{R}) \supset S^{n-1}= \lbrace [1,x_1,...,x_n] | x_1^2+...+x_n^2=1 \rbrace$$ and $B(0,1)= \lbrace [1,x_1,...,x_n] | x_1^2+...x_n^2<1\rbrace $. Show that ...
0
votes
2answers
38 views

Name for “bicontinuous function” that's not bijective?

We know that an invertible continuous function whose inverse is also continuous is called a homeomorphism. But is there a name for a not-necessarily-bijective function that is "bicontinuous" in the ...
1
vote
4answers
183 views

A Euclidean space that is homeomorphic to a non-Euclidean space

Is there a well-known example (preferably in dimensions 2 or higher) of two homeomorphic spaces: (1) a metric space with the Euclidean metric and (2) a metric space that is not Euclidean?
1
vote
2answers
68 views

Good book on Probability theory, Topology and Group theory for a beginner.

This year I will start three new 'branch' in mathematics : Probability theory, topology and group theory. I would like to know three complete books i.e. starting with the basics 'tools' whilst going ...
3
votes
1answer
27 views

Constructing a surjection from fundamental group of a mapping cone to Hawaiian Earring to $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$

If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X = $ (the Hawaiian Earring) which ...
2
votes
1answer
35 views

Relationship between Continuity and Countability

This is a consequence of one of the problems in elementary real analysis that I am attempting to solve. I have this doubt. Suppose $f$ is a continuous map from the reals to the reals. If the set ...
-2
votes
0answers
11 views

if I have a certain kind of metric on C([0, inf]), how to show it give the toplogy of uniform convergence on compacts?

I was give two kinds of metrics on C([0,inf]) and ask to show they give the topology of uniform convergence on compacts. Can anyone outline the procedure of the proof? Thanks.
1
vote
2answers
19 views

Disjoint compact subsets of a Hausdorff space are separated by disjoint open neighborhoods

Let $X$ be a Hausdorff space and let $A,B\subseteq X$ two compact subspaces which don't intersect. Show exist $U,V\subseteq X$ open which don't intersect s.t $A\subseteq U,B\subseteq V$. I ...
0
votes
0answers
28 views

Inner product spaces, normed spaces, metric spaces and topological spaces

I am collecting theorems or properties that hold in IPS, NS, MS or topological spaces, but not all of them. The reason is that I want to create some sort of overview over the respective spaces and ...
1
vote
1answer
18 views

Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
1
vote
2answers
47 views

Basis For A Topology

Let $X$ be a non empty set and $A$ is a subset of $X$. Show that the family of all subsets of $X$ which contains $A$, together with the empty set, forms a topology on $X$. (Use definition of basis ...
2
votes
0answers
57 views

Is it true that $\operatorname{Int}(A) \cap \operatorname{Int}(B) = \operatorname{Int}(A \cap B)$?

can someone please verify my proof? (a) Is it true that $\operatorname{Int}(A) \cap \operatorname{Int}(B) = \operatorname{Int}(A \cap B)$? (b) Is it true that $\bigcap ...
0
votes
2answers
34 views

If $U$ is open, is it true that $U = \operatorname{Int}(\overline{U})$?

Can someone please verify this proof? I am aware that there must be a similar question elsewhere, but I need help with my proof in particular. If $U$ is open, is it true that $U = ...
3
votes
2answers
76 views

Question about a basis for a topology vs the topology generated by a basis?

This is a really basic (no pun intended......no? Ok...) question about what it means to be a basis for a topology. Here is what I know: If $(X, \mathcal{T})$ is a topological space, and ...
0
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0answers
42 views

The definition of the basis of the topology

The definition of the basis elements of the topology says that if x belongs to the intersection of two basis $B_1,B_2$,then $\exists$ a basis element $B_3$ s.t. $x\in B_3$. I am considering why we ...
0
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0answers
20 views

Show that $\operatorname{Int}(A)$ and $\operatorname{Bd}(A)$ are disjoint, and $\overline{A} = \operatorname{Int}(A) \cup \operatorname{Bd}(A)$

Can someone please verify my proof? If $A \subseteq X$, we define the boundary of $A$ by $\operatorname{Bd}(A) = \overline{A} \cap \overline{X-A}$. $\operatorname{Int}(A)$ is defined as the ...
5
votes
2answers
43 views

Show that $\bigcup \overline{A_\alpha} \subseteq \overline{\bigcup A_\alpha}$

Can someone please verify my proof? I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Show that $$\bigcup \overline{A_\alpha} \subseteq ...
0
votes
2answers
37 views

Two Defs. of Neighborhood

Ask another question about Neighborhood(Ne): There are two definitions: http://en.wikipedia.org/wiki/Topological_space (Neighbourhoods definition) That is: (a) if N ∈ N(x), then x ∈ N (b) if N ∈ ...
1
vote
1answer
31 views

Intersection of an open dense set and any dense set is dense.

I am going through an introductory topology chapter and finding some difficulty in the following question. I had a chance to go through Daniel's and Brian's answers on this page. However, I am still a ...
1
vote
1answer
34 views

a example of two space that are homotopy equivalent but not homeomorphic

Is $D^2$ and the point space $P$ containing a point of $D^2$ homeomorphic? Are the two space of same homotopy type? I am seeking for a example of two space that are homotopy equivalent but not ...
1
vote
1answer
54 views

Prove $p_k\circ f$ continuous $\implies$ f is continuous

Let $X_1,\dots X_n$ topological space and $p_k:X_1\times\cdots X_n\to X_k$ the projection to the kth component. Let $Y$ be topological space and $f:Y\to X_1\times\cdots\times X_n$ function s.t ...
1
vote
3answers
68 views

Closure of the difference of two sets vs difference of their closures

This exercise is from Chapter 2, Section 17, number 8 (c), pag. 101, from Munkres's Topology. Let $A$, $B$ denote subsets of a space $X$. Determine whether the following equation holds: ...
0
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0answers
29 views

Def. about neighborhood on wiki

The def on wiki: "If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V \subseteq X$ that includes an open set $U$ containing $p$." And it says: "Note that ...
1
vote
1answer
67 views

Is the suspension of a countable collection of points in $\mathbb{R}$ a countable collection of circles?

I am extremely new to topology and taking an algebraic topology course, and I need some help understanding the behavior of suspensions. The problem I am working on asks about the suspension of the ...
0
votes
2answers
26 views

If closure of a set $D = \mathbb R^p$, then every element of $\mathbb R^p$ is a cluster point of $D$

Using the definition of closure of a set $A$ as the intersection of all closed sets in $\mathbb R^p$ containing $A$ , prove that If closure of a set $D = \mathbb R^p$, then every element of $\mathbb ...