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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3
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1answer
38 views

The open sets in Banach space

Let $X$ and $Y$ be two Banach spaces. And we set $X\times Y=\{(x, y): x\in X$ and $ y\in Y\}$. If we take a open set $U$ in $X\times Y$, then does $U$ has the form $U_{X}\times U_{Y}$? Here $U_{X}$ ...
5
votes
3answers
106 views

Why must an interior point of $E$ be an element of $E$?

This question takes place in a general metric space $X$. Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. This is like the ...
3
votes
1answer
40 views

Examples of connected $G_\delta$'s with empty interior that are not closed

Are there examples of non-empty connected $G_\delta$ sets in compact Hausdorff spaces that are neither closed or open and have an empty interior?
1
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1answer
65 views

Showing that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$

I have a question that asks me to show that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$ I have having trouble showing what I have is a quasi isometry. My map is simply: ...
1
vote
1answer
62 views

Show that topologies are the same

I just read a proof where it was said that if for each element in the topology 2 we find an element in topology 1 that is contained in this set and vice versa, then they are the same. How do I see ...
0
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1answer
38 views

Transforming a curve on an arc to a line

I have a function, actually a point cloud, (similar to a sine wave) on an arc with a known radius of curvature. I need to remove the curvature to regenerate the original function (or point cloud). ...
1
vote
1answer
26 views

Topologically equivalent metric spaces is an equivalence relation

I'm trying to prove that topological equivalence is an equivalence relation. Reflexivity was easy, and I'm sure transitivity is too, but I'm stuck on symmetry. My book's definition is that a metric ...
1
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1answer
16 views

Neighbourhood filter of an isolated point of a topological space

How can I prove this? Let $(X,\tau)$ a topological space, and $x \in X$. Then the neighbourhood filter $\mathcal{V}(x)$ is an ultrafilter if and only if $x$ is an isolated point of $X$ Thanks a ...
0
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1answer
18 views

finding accumulation points of a topology

in lipshutz book says I don't understand why the point $c$ isn't a limit point of $A$ since the open set $\{c,d\},\{a,c,d\},\{b,c,d,e\},X$ does contain a point of $A$ different from $c$
0
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1answer
19 views

Relation between dense subsets in the product map and dense subsets in each component

Let $\Omega$ bea Polish space and $X_1,\dots,X_n:\Omega\rightarrow\mathbb R^d$ be Borel measurable maps. Consider now the map $X:\Omega\rightarrow(\mathbb R^d)^n$ defined by ...
0
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1answer
39 views

How to find the topology of this subbase genarated?

I want to find topologies which uses this set as a subbase. two different questions says 1-X={a,b,c,d} and $\mathcal S $={{a,b},{b,c},{d}} 2- {[x,x+1]|$x\in$ R} in the first : when we take finite ...
0
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1answer
43 views

Finding the connected components of topological spaces

Find the connected components of the following sets: $(a) \; A=\{(x,y):y=\sin(1/x), x\in\mathbb{R}^+\},(b)\;A\cup\{(x,y):x=0,y\in[-1,1]\},(c)\;$The Cantor Set,$(d)\;\mathbb{N}$ with the cofinite ...
4
votes
1answer
46 views

A Polish space which is not locally compact

I want to find an example of Polish space which is not locally compact. I am thinking about the space of all continuous function from $[0,1]$ to $R$, endowed with the metric $d(f,g) = \sup_{x\in ...
1
vote
1answer
54 views

Is a function continuous iff its restriction to each element of an open cover is continuous

Let $(X;T_1)$ and $(Y;T_2)$ be topological spaces and let $A$ and $B$ be nonempty subsets of $X$ with $A\cup B= X$ Suppose $f:X\rightarrow Y$ is a function. Then prove or disprove: (a) if $f_A$ and ...
2
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2answers
95 views

Closed Sets and Open Sets

I have a few questions regarding open and closed sets. I am given a set: $$A = \left\{ \frac{1}{x}: x \in \mathbb{Z}^+ \right\},$$ I was asked to find the interior, closure, and boundary points. This ...
0
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0answers
36 views

Show that $(1,1,1,…)$ is a limit point of a set $A$.

Let $X_j$ be $\{0,1\}$, the 2 point set, with the discrete topology for $j = 1,2,…$. Let $X$ be the countable product of the $X_j$'s with the product topology. Let A be the set which consists of ...
0
votes
1answer
36 views

In a metric space, prove there is an invertible function $\Bbb R^n\to\Bbb R^n$ such that $f(a)=b$

I would like to prove the following theorem from Mendelson's Introduction to Topology: For each $a,b\in\Bbb R^n$, prove that there is a topological equivalence between $(\Bbb R^{n},d)$ and ...
1
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0answers
50 views

Is $ S^1$ a retract of $\mathbb R^2$? [duplicate]

I know that $S^1$ is a retract of $\mathbb R^2\setminus\{0\}$. After proving this as an exercise of a book by Munkres, we are asked to answer the following: "Is $S^1$ a retract of $\mathbb{R}^2$?" I ...
3
votes
1answer
57 views

Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
1
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1answer
47 views

is retract of a hausdorff space closed in that space?

If $Z$ is a topological space, we call $Y\subset Z$ a retract of $Z$ if there is a continuous map $r:Z \rightarrow Y$ such that $r(y)=y$ for all $y\in Y $. If $Z$ is Hausdorff and $Y$ a retract of ...
1
vote
1answer
40 views

$T_3$ is quasi isometric to $T_4$

I have a question which asks me to show that $T_3$ is quasi isometric to $T_4$, that is the three and 4 valence trees. I know that this means that I have to define a map $f:T_3\rightarrow T_4$ such ...
2
votes
0answers
60 views

How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
1
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1answer
35 views

Find a topological space such that $(A^a)^a \nsubseteq A^a$

I'm having some troubles with this problem, I hope you could help me: "Given $(X,\tau)$ a topological space, find $A \subseteq (X,\tau)$ such that $(A^a)^a \nsubseteq A^a$, where $A^a = \{ x \in A : ...
0
votes
1answer
49 views

Open subspace of a locally compact space is locally compact?

Open subspace of a locally compact space is locally compact? The definition of locally compact is given in Willard: A space X is locally compact iff each point in X has nhood base consisting of ...
0
votes
3answers
71 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 ...
8
votes
2answers
78 views

Topological distinguishibilty of $\infty$ after one point compactification?

Let $X$ be the one point compactification of some locally compact Hausdorff space. Let $\infty \in X$ represent the added point. Is there always a homomorphism $\phi:X \to X$ with $\phi: \infty ...
0
votes
0answers
30 views

Order topology on non discrete set

Set $X=\{1,2\} \times\mathbb Z^+$ where $\mathbb Z^+$ is positive intergers. Consider $X$ under dictionary order. The order topology on $X$ is not discrete. Why it is not discrete here? Why I cannot ...
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0answers
20 views

Separating convex sets in a tvs $X$.

I got doubt with the proof of this theorem. Let $X$ be a tvs, $A,B \subset X$ with $A$ an open convex set and $B$ convex such that $A \cap B = \emptyset$. Then there exists $f \in X^*$ (where ...
0
votes
0answers
25 views

Are all subbasis subsets of basis?

In topology, each element of basis $\{B_k\}$ can be expressed as finite intersections of elements of subbasis, i.e. $B_k=S_{n_1}\cap ...\cap S_{n_m}$ Does the meaning of "finite intersections" also ...
0
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0answers
34 views

Preimage of boundary

If $f$ is a continuous function between topological spaces, is it true that: $$f^{-1}(\partial A)=\partial f^{-1}(A)$$ for a subset $A$ of the domain? If not, what further requisites would I require?
0
votes
2answers
43 views

being completely normal implies perfectly normal?

i know that being perfectly normal implies being completely normal.but is the converse true?i've read somewhere that $\overline S_\omega$ is a counterexample but i don't know how to prove it. help ...
0
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3answers
83 views

Is d(0,a) = d(b,a+b) true in general?

As an exercise, I was trying to prove, given a distance metric $d$ on a metric space $X$, that if $a,b\in X$ then $d(0, a) = d(b, a+b)$ but I'm not seeing a way to do it. Is this necessarily true in ...
4
votes
1answer
81 views

Topological degree of a complex valued map defined over a circle

Given a continuous map $f \colon S^n \to S^n$, it induces a map $f_{*} \colon \tilde{H}_n(S^n) \to \tilde{H}_n(S^n)$ of the form $f_{*}(z)=k*z$, where $k$ is an integer. Define the degree of $f$ as ...
0
votes
1answer
41 views

Showing the space of Hermitian matrices is isomorphic to the Euclidean space.

The object in question is pretty straightforward: I would like to prove that the space of $N \times N$ Hermitian matrices, aka $\mathscr{H}_N$, is isomorphic to the Euclidean space, ...
0
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3answers
74 views

Prove that two paths on opposing corners of the unit square must cross.

I'm looking for a simple argument to the following: Given two (continuous) paths on the unit square, one from (0,0) to (1,1) and the other from (1,0) to (0,1), prove that the paths cross at some ...
7
votes
2answers
122 views

The dense topology

The definition of the dense topology confuses me. If $C$ is a category and $X \in C$, a sieve $S$ on $X$ is a covering for the dense topology iff for every $f : Y \to X$ there is some morphism $g : Z ...
1
vote
2answers
57 views

Constructing a sequence that is pointwise bounded but not uniformly bounded by points in a closed, nowhere dense set in $\mathbb{R}$.

I believe that this question below is asking for a sequence of functions that are bounded pointwise in $\mathbb{R}$ but NOT uniformly bounded in a closed, nowhere dense set of $\mathbb{R}$. Suppose ...
0
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0answers
39 views

Product topology of compact subsets

Suppose K,L are compact subsets of topological spaces X,Y respectively, and that KxL < W where W is open in XxY. Prove that for each x in K there exist sets U_x, V_x open in X, Y respectively and ...
0
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0answers
39 views

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous?

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous? I am trying to prove continuity by limits but am failing: suppose ...
3
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1answer
88 views

Open covers by simply connected sets and fundamental group

I have a set $X$ which is path connected and it have an open cover by sets $U$ and $V$ which are simply connected, I am looking for a reference that shows that $\pi_1(X)$ is the free group with number ...
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0answers
35 views

Is the closure of the cells in a CW complex compact?

As the cells in a CW complex afaik are homeomorphic to the open ball by definition, I was wondering whether this also means that their closure is a compact set? And if this is true, I would be ...
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votes
0answers
32 views

CW complex in $\mathbb{R}^n$?

I am supposed to show that a finite CW complex is homeomorphic to a compact subset of some $\mathbb{R}^n$, where $n$ is large enough. My idea was the following: A CW complex is also T3, which means ...
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0answers
29 views

Locally compact CW complex

I want to prove the folowing: A CW complex is locally compact iff every point has a neighbourhood that intersects with just finitely many cells. I already did the implication "locally compact", ...
1
vote
1answer
35 views

A Property equivalent to being a closed map

Please, I like you to view this statement and tell if I'm doing something wrong. Proposition: A function $f:X \rightarrow Y$ between topological spaces is closed if and only if for all $A \subseteq ...
3
votes
1answer
40 views

Show that $T$ is continuous with $\langle x,T(x)\rangle\geq 0$

Suppose that $T$ is a linear application on a real-Hilbert space $E$ such that $\langle x,T(x)\rangle\geq 0$ for all $x$. Show that $T$ is continuous. My attempt : We have for all ...
0
votes
1answer
24 views

Stating whether a space is complete [duplicate]

I'm asked to state whether or not $(X, d_u)$ and $(X, d_L)$ are complete where $d_u$ is the uniform metric and $d_L$ is the $L^1$ metric. All I need is to give the name of a supporting theorem or ...
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0answers
41 views

Open ball in infinite dimensional Banach space is not weakly open

I have to prove that open ball in infinite dimensional Banach space is not weakly open. I have no idea how can I do it. I think that I should reach contradiction with infinite dimensions.
0
votes
1answer
24 views

Set of limit points on a topological spaces.

I like to solve this problem. "Problem: In any topological space, the set of limit points of a sequence is closed." The proof is easy when we work with metric spaces, but how can I generalize this ...
1
vote
1answer
100 views

What is meant by gluing two metric spaces together?

"Gluing" constructions are common in topology: by gluing two disks along their boundaries we get a sphere; by gluing a cylindical "handle" to a sphere we get a torus, and so forth. If the original ...
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0answers
32 views

Subsets and finer filters

Suppose $G$ is a finer filter than $F$ in a topological space $X$. Is the net base in $G$ a subnet of the net base in $F$? I am using the definitions of General Topology of Willard. Thank you