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
82 views

Restriction of quotient map to open subset

Recently I encountered an alleged fact about restrictions of quotient maps, and I tried proving it. Arriving, with some help, at a proof sketch. But today I was told that the fact wasn't true, so I ...
0
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0answers
11 views

Does the quotient map need to take open sets to open sets, why? [duplicate]

The quotient map says that open sets in the image must be open in the preimage, but it says nothing about open sets in the domain needing to map to open sets in the image does it? Otherwise is this ...
1
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0answers
165 views

Prove the tangent space at a point $x$ of the $n$-sphere is the space $\{v \in \mathbb{R}^{n+1} : v\cdot x=0\}$

I can see why this is true but I'm not sure how to prove it, any help would be appreciated. Prove that the tangent space $TS^{n}_{x}$ at a point $x$ on the $n$-sphere $S^{n}:=\{x \in \mathbb{...
2
votes
2answers
56 views

How to calculate Euler characteristic of surfaces $K$ and $P$?

The book Introduction to Topology by C. Adams and R. Franzosa says : From the triangulations in Figure 14.8, we see that $\chi(S^2) = 2$, $\chi(T^2) = 0$, $\chi(K) = 0$ and $\chi(P) = 1$. And ...
1
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1answer
30 views

Finding open, disjoint sets

Let X be compact and Hausdorff. A, B be closed, disjoint subsets of X. Find U containing A, V containing B (U and V open and disjoint). My thought would be to fix x in A and pick y in B. By Hausdorff ...
3
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0answers
42 views

A problem of a discrete group of smooth isometries acting discontinuously on a smooth manifold.

Suppose that a smooth manifold $M$ is a metric space and that $\Gamma$ is a discrete group of smooth isometries acting discontinuously on $M$. Show that the action is necessarily properly ...
1
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1answer
119 views

Prove that $df_{x}: TM_{x} \rightarrow TN_{f(x)}$ is a well-defined map

I was wondering if someone could help me with the following problem, any help would be greatly appreciated. Let $f:M \rightarrow N$ be a $C^{\infty}$ map between smooth manifolds. Given $x \in M$, ...
0
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1answer
27 views

Intuition behind a Discrete and In-discrete Topology and Topologies in between

Looking around trying to find questions concerning the intuition behind discrete/indiscrete topologies, I haven't found much towards the essence of what these particular topologies imply about the ...
2
votes
1answer
274 views

Prove that the Cartesian product of two topological manifolds is a topological manifold.

I need help on the following problem, any responses would be greatly appreciated: Let $M$ be a topological $m$-manifold and $N$ be a topological $n$-manifold. Prove that $M \times N$ is a topological ...
9
votes
0answers
127 views

Norms and pointwise convergence

It is known that There is no norm $\|.\|$ on the space $E$ of continuous real-valued functions on an interval, say $[0,1]$ such that $f_n \to f$ for $\|.\|$ if and only if $f_n$ converges pointwise ...
1
vote
1answer
57 views

An open covering of the Topologist's sine curve.

How does an open covering of of the topologist's sine curve look like? I am asking since I want to show that it has a topological dimension 1.
5
votes
1answer
52 views

What is the closure of $S^1\times \mathbb{Z}$ in Homeo$(S^2\times \mathbb{R})$?

Let $X = S^2 \times \mathbb{R}$ and $G=S^1\times \mathbb{Z}$. Let $G$ act on $X$ as follows - $S^1$ acts on $S^2$ by rotation leaving the north and south poles fixed and acts trivially on $\mathbb{R}$....
2
votes
2answers
55 views

How to show that disjoint closed sets have disjoint open supersets?

Theorem: Let $S_0, S_1 \subseteq \mathbb{R}^n$ be disjoint and closed. Then there exist disjoint open sets $T_0, T_1 \subseteq \mathbb{R}^n$ such that $S_0 \subset T_0, S_1 \subset T_1$. How ...
0
votes
1answer
24 views

Countable image and a constant function

F:R to R has a countable image. Then f is a constant function. Prove this statement. I'm not really sure what it means a countable image and then really what to do with that. Any suggestions? Edit: ...
3
votes
6answers
347 views

General topology: looking for a brief and clear proof that the two main definitions of “closure” are the same.

There's a basic fact in general topology that I've never truly understood; namely, let $\mathbb{X}$ denote a topological space and $A$ denote a subset thereof. Then as I see it, there's basically two ...
3
votes
5answers
86 views

Definition of topology…“open in $X$” does it not involve a metric by definition?

I've been very confused with topology lately, and here's another one that I can't answer myself after spending hours staring at the definition. So, here's the definition I am using $X$ is a set. ...
1
vote
1answer
39 views

A specific problem on locally compact topological group Q and non existence of Haar measure

I have recently taken a course on topological groups and their Haar measure and I should first mention I am still a beginner so even though I thought about this for a while, I was hoping someone here ...
2
votes
2answers
46 views

Ultralimit of an eventually constant generalized sequence

Suppose that $x_{i \in I}$ is a generalized sequence on a compact Hausdorff space $X$, indexed by the directed set $I$, and with the property that $\exists \, j \in I$ such that $\forall i \geqslant j$...
0
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0answers
54 views

Can the Arzelà–Ascoli theorem be generalized for functions from a compact to a complete metric space?

Can the Arzelà–Ascoli theorem be generalized for functions from a compact metric space to a complete metric space?
0
votes
3answers
49 views

Is this a valid argument for Lipschitz Equivalence i.e. Topologically equivalent metrics?

Here, I saw the exact same question but I am trying to show it with Lipschitz Equivalence, for practice. The question is, A problem about topologically equivalent metrics but I will reiterate. I ...
2
votes
0answers
54 views

Is preimage of a point in a compact Hausdorff space under a continuous closed surjection compact?

If $X$ is compact Hausdorff and $p:X\rightarrow Y$ is continuous, closed, and surjective, can I say that for any $y\in Y$ that $p^{-1}(y)$ is compact in $X$. Ultimately I'm trying to prove that $Y$ ...
-2
votes
1answer
30 views

Is this space connected?

$F: [0,\infty) \to \mathbb{R}$ is defined by $F(x)=\sin(1/x)$ Is the graph of $F$ union $\{0\}\times[-1,1]$ connected?
25
votes
3answers
570 views

Connected, locally connected, path-connected but not locally path-connected subspace of the plane

I am looking for a set in the plane (with respect to the natural Euclidean topology) that is connected, locally connected, path-connected but not locally path-connected. I did not find one in Steen-...
1
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0answers
41 views

Small objects in TOP

Are spheres and disks small objects in TOP (more precisely, small with respect all continuos maps)?
0
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0answers
26 views

Box Topology and Regular

Determine whether $\Bbb R^\omega$ (Countable Cartesian product) in the box topology is Regular. This is one of the homework problems for my Introduction to Topology class. I started thinking about ...
2
votes
1answer
57 views

Questions about the C* subalgebra $h(\Gamma)$ of $l^{\infty}(\Gamma)$

Given $(\Gamma,d)$ a metric space that is discrete. We know that $l^{\infty}(\Gamma)$ is a commutative $C^*$ algebra that is isometrically isomorphic to $C(\beta\Gamma)$. Let $h(\Gamma)$ be a subset ...
2
votes
0answers
47 views

Octonionic Hopf bundle is a bundle.

We have the octonionic Hopf map $f: S^{15} \to S^8$, where $f$ is a bundle of fiber $S^7$. To me this is not obvious. How do I see that the octonionic Hopf bundle is indeed a bundle? Thanks.
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2answers
47 views

A collection of subsets of $\mathbb{R}$ that is not a topology

I want to show that the collection of sets of the form: $$\{(-\infty, x] : x \in \mathbb{R}\}$$ together with the empty set and $\mathbb{R}$, is not a topology for $\mathbb{R}$. But if I take the ...
0
votes
2answers
30 views

Are there any non constant - convergent sequence with discrete metric?

The sequence $\frac{1}{n}$ is convergent under euclidean metric. But not convergent with discrete metric. Is there a non-constant convergent sequence with discrete metric ?
1
vote
1answer
37 views

Convergence with respect to two topologies

I have an exercise in general topology as follows: Let $T$ and $T'$ be two topologies in a set $X$. What condition we need to put on these two topologies so that, if $(x_n)$ converges to $x$ with ...
1
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1answer
21 views

Contraction on continous functions with uniform norm

The problem is this: Let $G:C([0,1]) \rightarrow C([0,1])$ be defined by $$(Gx)(t) = \int_{0}^{t} sx(s) ds$$ $0 \leq t \leq 1$. I am supposed to show that $G$ is a contraction on $(C([0,1], d_{\infty}...
2
votes
2answers
44 views

If $U_1, U_2,…$ is an infinite collection of open sets , then their intersection is open ? True or false?

If $U_1, U_2,.....$ is a infinite collection of open sets , then their intersection is open ? True or false ? I proved that , If $U_1, U_2,......,U_n$ is a finite collection of open sets , then ...
2
votes
1answer
68 views

Baire's Theorem and Irrationals

I am asked to show that the irrational numbers are not a countable union of closed subsets of $\mathbb{R}$ given that if a complete metric space is the countable union of of closed subsets then at ...
2
votes
1answer
31 views

Example of homeomorphism of $S^1$ which behaves badly relatively to the Lebesgue measure

Could anyone give an example of a homeomorphism of $S^1$ which sends an open set of full Lebesgue measure on an open set which has not full measure ?
0
votes
1answer
48 views

prove that intersection of the family of the set is connected

I got no clue to solve this problem, because I can't find the connection between the compactness and the connectedness for the set family. Can anyone help me to solve this? I really appreciate. $X$...
0
votes
0answers
51 views

Find all interior, boundary, accumulation and isolated points of this set.

Let $S=\{ a\in\mathbb R\mid a \text{ is the root of a polynomial with integer coefficients}\}$. I'm looking to find all of the interior, boundary and accumulation point of $S$. I got that the ...
2
votes
0answers
101 views

Showing the Hausdorff metric inherits completeness

Let $(X,d)$ be a metric space and let $K(X)$ denote the set of all compact subsets of $X$. Then $(K(X), d_H)$ is a metric space, where $d_H$ is the Hausdorff metric. How can I show that if $X$ if ...
3
votes
2answers
156 views

Show that there is a invertible continuous function $h: \mathbb{Q} → \mathbb{Q}$ such that $h(−1) = 0$, $h(0) = 1$, $h(1) = −1$.

Show that there is a invertible continuous function $h: \mathbb{Q} → \mathbb{Q}$ such that $h(−1) = 0, h(0) = 1, h(1) = −1$. My attempt so far has been to try to split the rationals in [0,1] into ...
0
votes
3answers
108 views

Is this metric space complete?

Let $a, b \in \mathbb{R}$ such that $a < b$, and let $X$ be the metric space of all the (real- or complex-valued) functions defined and continuous on the closed interval $[a,b]$ with the metric $d$ ...
3
votes
2answers
44 views

Continuous open/closed surjection $p : \mathbb{R} \to \mathbb{Z}$

I am trying to solve the following problem. I have shown that the following collection of subsets of $\mathbb{Z}$ is a base for a topology $\tau$ on $\mathbb{Z}$: $\mathbb{B} =$ { {$2k + 1$} : $k \...
5
votes
4answers
164 views

Topology $\text{i})$ What is a topology? $\text{ii})$ What does a topology induced by a metric mean?

I am now trying to understand what a topology and a topological space is. Yes, I know the "formal" or "mathematical definition" of it, it is in my notes so it's easy for me to reiterate that. Please ...
0
votes
0answers
27 views

Proof that dimension of set is n

I have a Discrete Geometry question and I would really appreciate if someone could help me out with this. I have a set $K\subset \mathbb{R}^n$ s.t. $int(B_{n}) \subseteq K \subseteq B_{n}$ where $int(...
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0answers
32 views

$A\cup B$ and $A\cap B$ locally connected.

Let $X$ be locally connected, $X=A\cup B$, where $A,B$ are closed and $A\cap B$ is locally connected. Prove $A$ and $B$ are locally connected. Let $x\in A$. Then, either $x\in A\setminus B$ or $x\in ...
3
votes
2answers
79 views

Why is this not a well-defined $\Delta$-complex of the torus?

My lectures notes say that the second diagram isn't a well-defined $\Delta$-complex of the torus because the $2$-simplices aren't totally ordered. I don't really understand what that means. Let's ...
1
vote
1answer
25 views

A “Simple Chain of Regions” and Compactness in the Continuum

Let me just start by saying that I'm basically trying to prove this: How to prove every closed interval in R is compact? Except that I need to do it in a very strange way... I'm teaching an Inquiry-...
2
votes
2answers
52 views

$f: \mathbb{R}\rightarrow\mathbb{R}^2$ where $\overline{f(\mathbb{R})}\neq f(\mathbb{R})$

I'm having trouble coming up with a (continuous) function from $\mathbb{R}$ to $\mathbb{R}^2$ where the closure of the range (w.r.t. the standard topology on $\mathbb{R}^2$) is not the same as the ...
3
votes
1answer
54 views

Let $f:S^n\to S^n$ be continuous. Can the Image of An Open Set Under $f$ Have Non-Empty Interior?

Let $f:S^n\to S^n$ be a continuous map. Let $x\in S^n$ and $y=f(x)$. Assume that there is a neighborhood $U$ of $x$ in $S^n$ such that no element of $U-\{x\}$ maps to $y$. Question. Can the ...
0
votes
1answer
46 views

how to prove the commutativity of union and pullback without element?

Suppose F -> E and {U(i) -> E :a monomrphism} , how to prove the fibre product of F and the union of {U(i)} is the union of {the fibre products of F and U(i)} ? I think the union of {U(i) -> E} can ...
0
votes
1answer
54 views

Show a connected subset of $\mathbb R^2$

How can I show $\{(x_1,x_2)\in \mathbb R^2 :x_1^2+x_2^2=1\}$ is a connected subset of $\mathbb R^2$ MY SILLY ATTEMPTS INCOMING: Call the function set above E. Say I assume $f(E)$ is not connected in ...
-1
votes
1answer
66 views

Definition of the Cantor Function

What is the motivation for defining the cantor function $c(x)$ to be The value of $x$ in base $3$, with digits replaced and interpreted in base $2$? It seems very arbitrary. I can't understand why. ...