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; ...

learn more… | top users | synonyms (2)

0
votes
2answers
57 views

Prove Simply Connected

If $X = U \cup V$ with $U,V$ open and simply connected and $U \cap V$ is path connected, why is $X$ simply connected?
0
votes
1answer
15 views

How is $\pi : S^1 \times I \mapsto B^2$, defined as $\pi (x,t)=(1-t)x $ a closed map?

Consider the map $\pi : S^1 \times I \mapsto B^2$, defined as $\pi (x,t)=(1-t)x $ Here $B^2 $ is the closed unit ball in $\mathbb{R }^2 $, $S^1 $ is the unit circle. How is it a closed map? I ...
0
votes
1answer
34 views

Deformation retraction onto the boundary

If I have a square and I remove an open disc from its interior, there exists a deformation retraction onto its boundary. Is this also the case, if I remove a closed disc from its interior? Does the ...
3
votes
1answer
35 views

Image of Regular Map

Determine the image of the regular map $f: A^2 \to A^2$, $f(x,y)=(x,xy)$ and describe it from the point of view of topology. Would the image of f be $A^2$, because every point of $A^2$ is still in the ...
3
votes
1answer
43 views

How to show that $\mathbb R$ is not Rothberger, and how to show that it is not Menger?

We say that topological space $X$ is Rothberger $S_1(\mathcal O,\mathcal O)$, if: For any sequence of open covers of $X$, $\{ \mathcal U_n | n \in \omega \}$, one can always find a sequence $\{ ...
1
vote
1answer
37 views

Proving that $f(\bar Z)\subset\overline {f(Z)}$ when $f$ is a continuous map

I'm trying to solve this question from my textbook: Let $f:X\rightarrow Y$ be a continuous map and let $Z \subset X$. Prove the inclusion $f(\bar Z)\subset\overline {f(Z)}$. Thanks in advance ...
-1
votes
1answer
53 views

Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
0
votes
1answer
33 views

describing all possible topologies on a set

Assuming the following: let $X$ be a set with two elements, $X$ = {$a,b$}. what are all the possible topologies on $X$ ? The answer I've come up with is: $$\tau_{T}=\{ \emptyset, X\} $$ $$ ...
1
vote
3answers
60 views

topology homework question

so I got this question for homework: let $x$ be a topological space and let $A \subset C$. one sets $\alpha(A) = \mathrm{Int}(\bar{A})$, and $\beta(A) = \overline{\mathrm{Int}(A)}$. Prove if $A$ is ...
0
votes
1answer
14 views

maximal independent set in a graph

Let $G$ be a graph and $A$ is a subset of vertex set of $G$. $A$ is said to be independent if for any $x, y \in A$, $(x,y) \notin E(G)$, i.e $x$ and $y$ not connected by an edge. Further A is said to ...
0
votes
4answers
76 views

Continuous bijective function between $\Bbb{R}$ and $[0,1]$?

Does a continuous bijective function from $\Bbb{R}$ to $[0,1]$ exist? If not please explain. Here $[0,1]$ and $\Bbb{R}$ have the usual topology.
2
votes
0answers
44 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
4
votes
0answers
181 views

Questions related to intersections of open sets and Baire spaces

I was curious a little about the classes of topological spaces described below. For each of them I want to ask whether they have been studied and whether they have some interesting properties. ...
1
vote
0answers
22 views

If every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space

Show that if every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space. (Munkres "Topology", 48.3) Here is what I tried : Let $\{U_n\}_{n \geq 1}$ be a collection of ...
2
votes
0answers
62 views

Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire? [migrated]

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
4
votes
2answers
60 views

half space is not homeomorphic to euclidean space

We define, half space $H^n$ = $\{(x_1,x_2,...,x_n) | x_n \geq 0\}$. Can anyone suggest, how to prove that $H^n$ is not homeomorphic to the euclidean space $R^n$.
0
votes
2answers
39 views

Is this map a quotient map?

Let $X,Y$ be topological spaces and $\pi: X \to Y$ the quotient map generated by $\tau_Y$, the topology of $Y$. $U \subset X$ is an open or closed subset, saturated respect to $\pi$ (i.e. ...
1
vote
0answers
25 views

Proof of Euler Characteristic for Sphere

Theorem 1. All cell decompositions of a sphere $S$ have Euler characteristic 2. This is well-known, but I had this idea for an intuitive proof: for any cell decomposition $\Gamma$ with $V$ ...
0
votes
0answers
18 views

What about Alexandroff Duplicate? [duplicate]

Let X be any topological space and let A(X)=X∪X′ such that X prime= X union {1} be its Alexandroff Duplicate. If X is normal, is A(X) also normal? Thanks for any help.
-2
votes
0answers
31 views

Real analysis and topology

Let $(X,\tau)$ be Hausdorff topological spaces, Show that $\tau$ is a semi-ring if and only if $\tau$ is the discrete topology.
5
votes
1answer
87 views

Struggling with Topology. Any advice?

I'm a Junior Mathematics major at a small Liberal Arts college. I'm currently taking first semester topology (Munkres text). I feel like I'm barely able to tread water in this course. I was able to ...
0
votes
2answers
66 views

Can you take off a sweater while wearing headphones?

This seems like a graph theory problem, but I'm not sure how to approach it. To clarify potential ambiguities, let's set up the situation. You are wearing a sweater (with one arm through each ...
0
votes
2answers
23 views

Compact subset of a closed subspace: compact in the whole space?

Imagine that you have a topological space $(X,\tau_{X})$ and a closed subset $Y$. Say that within $Y$ we have a subset $K$ that is compact in the subspace topology $\tau_Y$. Is $K$ compact in ...
1
vote
1answer
27 views

Misunderstanding of an exercise on topology?

Let $\mathbb{R}$ be the set of reals with a topology $\mathcal{T}$ such, that each $x\in\mathbb{R}$ has a base of open regions ...
0
votes
1answer
39 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
0
votes
1answer
27 views

Is Alexandroff Duplicate normal?

Let $X$ be any topological space and let $A(X)= X\cup X'$ be its Alexandroff Duplicate. If $X$ is normal, is $A(X)$ also normal? Thanks for any help.
1
vote
1answer
13 views

Topological subspaces with order

$Y\subset (X, )$ is said convex if for all $a, b \in Y$ and with $a<b$ where interval $(a, b) \in Y$. Show that in this case the topologies T $(\mathbb{R}^2,\mathcal{T}^{\trianglelefteq })$ ...
1
vote
1answer
19 views

Need help on Quotient Spaces

Definition: Suppose $X, \tau$ is a topological space and $R$ is an equivalence relation on $X$. Let $X/R$ denote the set of $R$-equivalence classes. Define the function $f$ from $X$ to $X/R$ by $f(x) ...
0
votes
1answer
48 views

Why is this map closed?

I've encountered with this question during reading J.M.Lee's book: Define an equivalence relation on $\mathbb{R}^{2}$: $(x,y)\sim(x',y')$ iff $(x',y')=(x+n,(-1)^{n}y) $ for some $n\in\mathbb{Z}$ ...
5
votes
1answer
42 views

Is this condition sufficient to determine the linear space is of finite dimension?

From the Banach theory we knew that: 1) A linear space(a vector space endowed with its vector topology) $X$ of finite dimesion $dimX=n$ has the following property: If ${\left\| \bullet \right\|_1}$ ...
3
votes
3answers
43 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
0
votes
0answers
15 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
1
vote
1answer
16 views

Dealing with quotient space homeomorphism.

Let $\mathbb{R}^2$ be the plane with the Pythagorean topology. Let $E$ be the equivalence relation defined by the partition $\{\{(x,y) | y < 0\}, \{(x,y)|y \geq 0 \}\}$. Here, $\{(x,y) | y < ...
1
vote
1answer
28 views

Confirm basis for $X \times Y$ topologies

$X = \{a, b\}$ and $\tau = \{ \emptyset, \{a\}, X\}$ $Y = \{c, d, e\}$ and $\tau' = \{ \emptyset, \{d\}, \{e\}, \{d, e\}, Y\}$ Now, a basis for the topology of $X \times Y$ would be: $\{\emptyset, ...
1
vote
0answers
45 views

An example of a topological space which is a group, but is not a “topological group”

Is there any example of a topological space $X$ which has a group structure, but the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are not continuous?
0
votes
0answers
13 views

What's an example of a Quotient/Identification Space for this topological space?

I haven't been able to find an example with numbers anywhere on the internet and was hoping someone could help. If I have $X = \{1, 2, 3\}, \tau = \{\emptyset, \{1\}, \{2\}, \{1, 2\}, X\}$ as my ...
1
vote
3answers
44 views

Why is $[0,1]$ not homeomorphic to $[0,1]^2$?

Why is $[0,1]$ not homeomorphic to $[0,1]^2$? It seems that the easiest way to show this is to find some inconsistency between the open set structures of the two. It is clear that the two share the ...
1
vote
1answer
18 views

Continuous functions with trivial topology

Definition: Let $X, \tau$ and $Y, \tau'$ be topological spaces. Then a function $f$ from $X$ to $Y$ is said to be continuous if given any open subset $U$ of $Y$ then $f^{-1}(U)$ is an open subset of ...
4
votes
2answers
59 views

Different definitions of P-Points (ultrafilters)

Recently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...
1
vote
1answer
21 views

Intuition behind why lines and points are closed sets in $\mathbb{R}^2$

So, a subset, $U$, of $\mathbb{R}^2$ is open if we can pick any point, $x \in U$ such that for some real number $\epsilon > 0$, given any point $y \in \mathbb{R}^2$ a distance less than $\epsilon$ ...
2
votes
2answers
187 views

Help with Proof explanation

I need help in understanding a (topological) proof of Fundamental theorem of algebra. Here is the Proof: Suppose $f(z)=a_nz^n+...+a_0$ with $a_0 \neq 0, n \geq1.$ WLOG, assume that $a_n=1.$ We ...
0
votes
1answer
35 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
0
votes
1answer
26 views

Can we take the infimum over a variable set?

Suppose that we have a family of functions $\lbrace f_{\alpha}(x)\rbrace_{\alpha}$ define on an open set of $\mathbb{R}^{m}$, and $\alpha$ runs over a set $\Gamma$. Assume that the family is ...
0
votes
1answer
13 views

Finding the boundary of a set

Let $I$ be any interval in the real line and consider the set $A = I \cap \mathbb{Q} $.Notice $A \subseteq \mathbb{Q} $. then $A$ must have measure zero since $\mathbb{Q}$ does. Is the boundary of $A$ ...
3
votes
1answer
38 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
1
vote
2answers
52 views

What is this quotient space of the torus?

Suppose we have a $\mathbb{Z}/2\mathbb{Z}$ action on torus $\mathbb{T} \times \mathbb{T}$ by $(\xi,\theta)$ goes to $(-\xi,\bar{\theta})$. Then what is the quotient space?
3
votes
1answer
42 views

About the Stone-Čech universal property

There's something I am missing here and I dont' know what it is. I understand that the Stone-Čech compactification of $X$ satisfies the property that for every continuous map $f: X \rightarrow K$ ...
1
vote
1answer
32 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
0
votes
2answers
29 views

Intersections of open and closed sets

How do I show that if $U$ is open in $X$ and $A$ is closed in $X$, then $U-A$ is open in $X$, and $A-U$ is closed in $X$? So far I have that $X-A$ and $U$ are open, hence $U \cap (X-A)$ is open. I ...
2
votes
0answers
42 views

What does base point by us for algebraic topology?

This may be a vague quesion. I am confusing between base pointed case and non base pointed case in algebraic topology. Is there any convinience in base pointed case? For example, it leads to the ...