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
141 views

Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group?

And what else can be said, if so? In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (It also has a two-sided ...
2
votes
2answers
338 views

proving something about the infimum distance between 2 set

Let $S,T$ be 2 set. Prove that there is a $x\in S$ s.t, $d(x,T)=d(S,T)$ if $S$ is a compact set. Here, $d(S,T)$ denoted the $\inf\{d(s,t):s\in S, t\in T\}$.
10
votes
3answers
535 views

What are the epimorphisms in the category of Hausdorff spaces?

It appears to be the case that the epimorphisms in $\text{Haus}$ are precisely the maps with dense image. This is claimed in various places, but a comment on my blog has made me doubt the source I got ...
3
votes
2answers
98 views

Closed bounded subset of $\mathbb{R}$.

Let $X$ be a closed and bounded subset of $\mathbb{R}$. Is it true that $X$ is a finite union of closed intervals in $\mathbb{R}$? (*) I think that if we choose $X$ is a Cantor set, then $X$ ...
3
votes
2answers
218 views

Paradox as to Measure of Countable Dense Subsets?

Consider the set $E=\mathbb{Q}\cap[0,1]$, and let $\{q_{j}\}_{j=1}^{\infty}$ be some enumeration of this countable set. For every $\epsilon>0$, the cubes $\{Q_{j}\}_{j=1}^{\infty}$ of length ...
1
vote
0answers
45 views

Manifolds question

Let $M$ subset of $R^{n+p}$ be the zero set of a $C^\infty$ mapping $g:R^{n+p} \rightarrow R^{p}$. Assume that the Jacobi matrix of $g$ has rank $p$ everywhere on $M$. Show that $M$ is an ...
0
votes
1answer
119 views

Complement of deformation retract

Let $X$ be a topological space and $V$ and $N$ are subspaces of $X$ such that $N$ deformation retracts onto $V$. I want to show that $X-V$ deformation retracts onto $X-N$. So i need to construct a ...
1
vote
1answer
282 views

A wedge sum of circles without the gluing point is not path connected

I am trying to prove that a wedge sum of two circles is not a topological manifold, to do so I am showing that the wedge sum without the gluing point is not path connected while the $\mathbb R^2$ ...
2
votes
2answers
411 views

Retraction map from unit disk to its boundary

Given two continuous surjective functions $f$ and $g$ from the unit disk to itself and $f(z) \neq g(z)$ for all $z$ in the unit disk is it possible to construct a retraction map from the unit disk to ...
1
vote
2answers
266 views

Simply connected

Is it possible for the closure of a simply connected domain in the complex plane to not be simply connected? Intuitively it seems the closure is simply connected but I can't prove it. Is it enough to ...
0
votes
2answers
376 views

Heine-Borel covering property (proof by contradiction)

This is taken from Stein & Shakarchi's Real Analysis, page 3. "Assume $E$ is compact, $E\subset\bigcup_\alpha \mathcal{O}_\alpha$, and each $\mathcal{O}_\alpha$ is open. Then there are finitely ...
1
vote
1answer
341 views

Why completion of a metric space $X$ is 'unique' upto isometry?

Let $(X,d)$ be a metric space. Let $({X_1}^*, {d_1}^*)$ and $({X_2}^*, {d_2}^*)$ be completions of $(X,d)$ such that $\phi_1:X\rightarrow {X_1}^*$ and $\phi_2:X\rightarrow {X_2}^*$ are isometries. ...
2
votes
0answers
96 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
7
votes
3answers
3k views

A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...
1
vote
4answers
211 views

Proof that $\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$

Given a topological space ($X$, $\mathcal{T}$) and the following defintions $\partial A$ := { $x$ | every neighbourhood of $x$ contains points from $A$ and from $X \setminus A$ } $\overline{A}$ := ...
0
votes
0answers
66 views

First examples in triangulations

I am starting to study about triangulations in my algebraic topology course. We have seen the triangulation of the sphere, the closed disc and so on. Intuitively it's ok, however I couldn't find any ...
2
votes
4answers
235 views

Proof that $(a,b)\not\cong[a,b]$

How to prove that $$(a,b)\not\cong[a,b]$$ (not homeomorphic) as subsets of real line? Is it true that in some topology $(a,b)$ is closed? Thanks a lot!
2
votes
1answer
99 views

How do I prove that a map from a quotient space is continuous

I'm taking some maps from quotient spaces to prove the continuity. I'm thinking if I can use the characteristic property to prove the continuity of the map. For example, take the map $f:\mathbb R/\sim ...
0
votes
1answer
35 views

Proving that $\partial P'\subset \partial P $ if and only in $P'\cap P^0 \subset P'^0$

As the topics. Proving that $\partial P'\subset \partial P $ if and only in $P'\cap P^0 \subset P'^0$. I am not sure how to start
0
votes
2answers
2k views

How to find limit points for this set?

Given a set $$S = \left\{\dfrac{1}{a} + \dfrac{1}{b} : \text{ where } a, b \in \mathbf{N}\right\}\;.$$ How could I find the limit points of this set? My idea is to consider as $a \rightarrow ...
7
votes
4answers
206 views

How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
0
votes
2answers
97 views

Does a compact subspace have to be closed in an arbitrary metric space?

For Euclidean spaces, we have that a compact subspace has to be closed (and bounded.) But how about an arbitrary metric space? Or how about an arbitrary topology space?
6
votes
4answers
139 views

It is possible to prove that these two collections generate the same topology on $ \mathbb{X} $?

Let $ \mathbb{Y} $ a topological space whose topology $ \tau $ is generated by a collection of subsets $ \mathcal{B} \subset 2^\mathbb{Y} $. In other words the topology $ \tau $ of $ \mathbb {Y} $ is ...
1
vote
0answers
104 views

The complement of a submanifold in a manifold

Let $X$ be a topological $n$-manifold and $N$ a $d$-submanifold of $X$, ($d\leq n$), then under what conditions on $X$ and $N$ do we have that the complement $X-N$ is again a manifold and what is the ...
1
vote
2answers
170 views

the geometric boundary of a manifold

Let $X$ be a topological manifold with geometric boundary $\partial_g X$ (here the subscript $g$ to indicate geometric boundary which is a different notion from topological boundary $\partial_t X$). ...
2
votes
1answer
146 views

set of all trace $1$ matrices are connected?

the heading is the question and here are my two approach, I want to know are they correct or not, if not I need to know the answer 1) They are path connected as $\gamma(t)=At+(1-t)B, t\in [0,1]$ ...
4
votes
2answers
112 views

open subsets in topological groups

I'm starting to study topological groups, and I noticed that Every single theorem in topological groups I have to use the following statement: Let $G$ be a topological group and U an open subset of ...
0
votes
1answer
186 views

Is there a continuous surjective from $S^1$ to $[0,1]$?

Is there a continuous surjective map from $S^1$ to $[0,1]$?
4
votes
1answer
140 views

Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
0
votes
1answer
313 views

Does a piecewise-continuous function need to be defined at its points of discontinuities?

Is the following function considered piecewise-continuous?? I'm reading conflcting definitions in different places: some highlight that that the function need not be defined at the (jump/removable) ...
2
votes
2answers
145 views

Diffeomorphism in Euclidean spaces

Let $a,b \in R^n$. Show that there is a diffeomorphism of $R^n$ carrying $a$ to $b$ which is the identity outside of an open set. I think I have a proof of this using integral curves but I would ...
4
votes
1answer
73 views

Showing closure in the $\|\cdot\|_1$ norm

Let $X=C[0,1]$ and $W=\{f\in X\mid f(0)=0\}$. What is the closure of $W$ wrt the 1-norm $\|\cdot\|_1$. My solution is as follows: The closure is the whole space $X$. To see this take any ...
1
vote
1answer
228 views

$0$-dimensional and $G^{⋆⋆}$-regular

1 ) Why $X = \{ 0 \} \cup \{ 1 / n : n \in \mathbb{N} \}$ is $0$-dimensional ? 2 ) Let $X$ be a space and $G$ a topological group, Why If $X$ is $0$-dimensional in the sense of ind, then $X$ is ...
2
votes
1answer
60 views

question regarding metric spaces

let X be the surface of the earth for any two points on the earth surface. let d(a,b) be the least time needed to travel from a to b.is this the metric on X? kindly explain each step and logic, ...
0
votes
0answers
81 views

a question on Hopf fibration

While reading a note on hopf fibration we came proving $\mathbb{CP}^1$ is homeomorphic to $\hat{\mathbb{C}}$ author says like $\mathbb{CP}^1$ has the quotient topology $\mathbb{CP}^1=S^3/\sim$. We ...
4
votes
1answer
82 views

Gluing axiom of a TQFT

In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT. When they come to prove the gluing axiom, they just mention that "...This statement is ...
1
vote
3answers
419 views

How are a set of matrices a topological space?

For instance, is the topology on the set of all $2 \times 2$ real matrices basically $\mathbb{R}^4$
2
votes
3answers
121 views

Metric Spaces, determining whether a point is in the interior, boundary, or interior of the complement.

I'm really stuck on this Real Analysis problem, if anyone would mind helping me. Let $(X,d)$ be a metric space, and let $A$ be a non-empty subset of $X$. How do you show whether any point $x \in X$ ...
14
votes
3answers
675 views

Cantor set and countability.

The Cantor set is closed, so its complement is open. So the complement can be written as a countable union of disjoint open intervals. Why can we not just enumerate all endpoints of the countably ...
0
votes
1answer
81 views

Topological Equivalence between $(- \pi/2, \pi/2)$ to $R$

I know that the key is to use $\tan$ and $\arctan$ to do it. Take any $(a,b) \subset \mathbb{R}$, $(a,b)$ is open. Now I want to show $\tan^{-1}(a,b)$ is open. I need a hint for the next step (just a ...
3
votes
1answer
85 views

What does this free quotient space look like?

Let $S^2=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2=1\}$ and $S^1=\{(s,t)\in \mathbb{R}^2|s^2+t^2=1\}$. Suppose that $\mathbb{Z}/2\mathbb{Z}$ acts on $S^2\times S^1$ in such a way that the generator of ...
0
votes
1answer
331 views

Prove that the limit points of open interval $A=(2,3)$ are all the points of the interval $(2,3)$.

Prove that the limit points of open interval $A=(2,3)$ subsets of the real numbers, are all the points of the interval $(2,3)$ including $2,3$. This is my solution (sketch). I consider a ...
0
votes
1answer
96 views

Compact and sequentially compact in $\Bbb R$ and $\Bbb R^n$

I read that in a metric space compactness and sequential compactness mean the same thing. In $\Bbb R$ is sequential compactness equivalent to compactness? I see some definitions of Heine–Borel theorem ...
1
vote
1answer
203 views

Sequence of Functions in the box and product topologies.

All topologies mentioned are on the cartesian product $\mathbb{R}^{[0,1]}$. The sequence of functions $(f_k)$, for which, $f_k(x) = x^k, x\in [0,1]$, converges pointwise but not uniformly to $f(x)=\ ...
1
vote
1answer
81 views

Some properties of Polish space

Let $X$ be a separable complete metric space. I wonder if following properties hold in ZF. Limit Compact ⇒ Compact Does there exist a function$f$ such that $f(E)$ is closed and $f(E)\subset E$, for ...
2
votes
1answer
130 views

How do I prove that $\mathbb CP^n$ is a 2n-manifold?

I'm struggling to prove that $\mathbb CP^n$ is 2n-manifold. We can defined the $\mathbb CP^n$ as the equivalence relation $(z_1,z_1,...,z_{n+1})\sim(w_1,w_1,...,w_{n+1})$ iff $z_i=\lambda w_i$, ...
0
votes
2answers
214 views

Limit Points within a Set

If I have an uncountable subset $A \in \mathbb{R}$, and we assume A is nonempty, does it follow that every point within $A$ is a limit point of $A$ from the density of $\mathbb{Q}$ in $\mathbb{R}$ ...
2
votes
2answers
91 views

Discrete property

Why $\mathbb{Z}$ with $p$-adic topology is non-discrete? Note1 : discrete : each singleton is an open set. Note2 : Let the topology $\tau$ on $\mathbb{Z}_p$ be defined by taking as a basis all sets ...
0
votes
3answers
159 views

Continuous map on topologial space with trivial topology

This is the problem: I have a space $X=\{a,b\}$ with trivial topology (open sets are empty set and whole $X$) and continuous map $f\colon X\to Y$. ($Y$ is arbitrary space). Can I conclude that ...
3
votes
1answer
233 views

Sorgenfrey plane is not normal = Help understanding the proof

So I'm looking at this proof, which is presented as a problem in Gamelin & Greene but I'm having some trouble understanding it. ...