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)

1
vote
0answers
20 views

A subset E of $R^n$ is totally bounded if and only if E is bounded

I am studying Compactness in metric space with Gamelin and Greene's Introduction to Topology and am confused about lemma 5.4 in the book. A metric space $X$ is totally bounded if for each $e > 0$, ...
3
votes
1answer
55 views

Is there an injective continuous map $\mathbb{R}^2 \rightarrow \mathbb{R}$?

It is commonly known fact that there exists a continuous surjective map $\mathbb{R} \rightarrow \mathbb{R}^2$. So it bids to ask: Is there an injective continuous map $\mathbb{R}^2 \rightarrow ...
0
votes
6answers
39 views

To show a set is open

Given $A \in \mathbb{R}$ be open define $B = \lbrace{(x,y) \in \mathbb{R}^2 : x \in A} \rbrace$ Show that $B$ is open in $\mathbb{R}^2$
1
vote
0answers
9 views

How to show that every Suslin tree is Frechet-Urysohn

A topological space $X$, is Frechet-Urysohn, if, given $x \in \overline A \subset X$, there exists a sequence of points in $A$ which converges to $x$. I am trying to prove that every Suslin tree, is ...
0
votes
0answers
9 views

is there any relation between category theory and differentiation theory in locally convex spaces

Is there any relation between category theory and differentiation theory in locally convex spaces. If it so , what type of relation.can you conclude in one or two lines
1
vote
0answers
28 views

Sequence of increasing compact sets

Suppose $X$ is a locally compact metric space which is $\sigma$-compact. Let $K$ be a compact subset of $X$. We can find a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + ...
0
votes
1answer
19 views

Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...
6
votes
2answers
67 views

Cantor's Teepee is Totally Disconnected

Let $C^\prime$ be the Cantor set and let $C = C^\prime \times \{0\}$ (viewed as a subset of $\mathbb{R}^2$). For $c \in C$, let $L(c)$ denote the half-closed line segment connecting $(c,0)$ to ...
1
vote
2answers
38 views

Give an example of a non-separable subspace of a separable space

I'm trying to find an example of a non-separable subspace of a separable space. What kind of examples are there?
1
vote
1answer
22 views

Existence of a neighbourhood of a compact set ( from james fibrewise topology)

I'm reading James' Fibrewise topology book and I'm trying to understand the proof of proposition 7.4 , it says: Let X be a proper G-space . Then X is fibrewise regular over X/G. Proof For any $x \in ...
2
votes
1answer
35 views

The set $S=\{(x,y) \in \mathbb{R}^{n} \times \mathbb{R}^n = \mathbb{R}^{2n} ; x \neq y\}$ is connected if $n \geq 2$.

When n = 1 it is easy to see that is not connected, it just take the split open $ S=U_1 \cup U_2$ such that $U_1 = \{(x,y) \in \mathbb{R}^2 ; x > y\}$ is $U_2 = \{(x,y) \in \mathbb{R}^2 ; x < ...
3
votes
0answers
36 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
1
vote
1answer
47 views

Show that $f(\operatorname{int} A) = \operatorname{int}(f(A))$ when f is a homeomorphism.

Show that $f(\operatorname{int} A) = \operatorname{int}(f(A))$ when f is a homeomorphism. My trial is as follows. It seems to me that there is an error or it needs more detail. Any answer or ...
0
votes
1answer
11 views

In metric space, f:homeomorphism. About preimage of an open ball

In metric space, let f be a homeomorphism. Is the preimage of an open ball still open ball? I know that the preimage of an open set is open. So I can take open neighborhood of a point. But in my ...
0
votes
1answer
26 views

Let $X$ and $Y$ be topological spaces and let $f: X\to Y$ be an onto continuous map. If $X$ is separable then $Y$ is also separable.

Let $X$ and $Y$ be topological spaces and let $f: X\to Y$ be an onto continuous map. If $X$ is separable then $Y$ is also separable. To prove this, let $A$ be a countable dense subset of $X$. Then by ...
2
votes
1answer
22 views

Cylinder with bases collapsed to a point.

The problem, although arising from some deeper facts, is quite simple. I would like to visualise the quotient space $A$ given by the cylinder $I\times S^{1}$ ($S^{1}$ is the circle in $\mathbb{R}^{2}$ ...
1
vote
0answers
63 views

$\ f \colon X \to X $ ,continuous function where X is compact,Hausdorff space.Show $\exists A$ st $f(A) =A$.

Suppose $\ f \colon X \to X $ is a continuous function from a compact,Hausdorff space to itself. Prove that there exists a subspace $A$ such that $f(A) =A$. I came up with an answer based on nets ...
4
votes
2answers
69 views

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
0
votes
1answer
22 views

Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property

Let $X$ be a Hausdorff topological vector space. Let $C$ be a nonempty compact subset of $X$ and $\{C_\alpha\}_{\alpha \in I}$ be a collection of closed subsets such that $C_\alpha \subset C$ for each ...
0
votes
0answers
19 views

Prove A is compact? [duplicate]

If A is infinite subset of (N,F) Cartesian product (R,F) where F is a Fort space topology and a particular point is (1,1). How can I prove that A is compact if A contained (1,1)?and prove that A is ...
-3
votes
1answer
54 views

How can I prove that A is compact? [on hold]

If A is infinite subset of (N,F) Cartesian product (R,F) where F is a Fort space topology and a particular point is (1,1). How can I prove that A is compact if A contained (1,1)? And prove that A is ...
1
vote
1answer
22 views

Existence of Nested Countable Neighborhood Basis

I'm stuck on an intro to point-set topology homework problem. I'm given that $(X, \mathcal T)$ is a topological space and $p\in X$ has a countable neighborhood basis. I need to then show that $p$ has ...
0
votes
0answers
54 views

Show that $f(\bar A) \implies \overline{f(A)}$.

Def) X:a metric space, $Y\subset X$: a subset. A point $x\in X$ is adherent to Y if $B(x;r) \cap Y \neq \emptyset \quad \forall r > 0.$ Def) $\bar Y := \{x\in X \mid x \text{ is adherent to } Y\}$ ...
2
votes
1answer
38 views

Proving a injectivity in a separable Hausdorff space.

I was just reading through a proof about the maximum possible cardinality of a separable Hausdorff space, but I'm stuck on one part of it. The essence of the proof is copied and pasted below, and they ...
0
votes
1answer
30 views

find a homotopy between $a^{-1}*a$ and $k$

$k$ is the identity element in the fundamental group, i.e $\forall a\in L(X,x_0), a*k=k*a=a$ Note that $a^{-1} (r)=a(1-r)$ so $$(a^{-1}*a)(r)=\left\{ \begin{array}{ll} a(2(1-r)-1) & ...
1
vote
1answer
29 views

Show that any infinite set $X$ may be endowed by a metric d such that $X$ has a limit point in $(X,d)$

This is an exercise I've been dealing with for a few days; I was wondering if anyone could help me with a hint or just telling me the answer. Regards
0
votes
1answer
43 views

Signed finite Radon measures with vague topology

If $X$ is a locally compact and $\sigma$-compact metric space. Let $M(X)$ be the space of signed finite Radon measures on $X$. (1) Show that measures with finite support is dense is $M(X)$ in the ...
2
votes
1answer
44 views

Does $\{A={\rm core}(A)\}$ form a topology?

I've the following question: If $V$ be an arbitrary (real) vector space and if for any subset $A\subset V$ we denote by $ {\rm core}(A)$ the set $\{a\in A\> |\> \forall v\in V \exists T>0 ...
2
votes
1answer
56 views

What is $\mathbb R^\omega$?

I have seen $\mathbb R^\omega$ mentioned in my topology texts but cannot find where $\omega$ is defined. Could someone please tell me what it means in comparison to $\mathbb R^n$?
0
votes
2answers
41 views

basic question of topology involving compactness and convexity

Consider in $R^n$ a compact and convex set $A$ with $int(A) $ nonempty. then $\overline{int(A)} = A$ ?. i have no idea to prove this. In this direction i only know the following (and hard to ...
1
vote
1answer
40 views

Fundamental Group of Klein Bottle?

Let $C^{*}=C \setminus \{ 0 \}$. What is the fundamental group of $C^{*}/H$, here $H=\{\psi^n;n \in \mathbb{Z}\}$ with $\psi(z) = 2 \bar{z}$?
0
votes
3answers
30 views

X :compact and continuous function $f(x)\neq x$

Let (X,d) be a compact metric space and $f:X\to X$ be a continuous function such that $f(x)\neq x,\: \forall x\in X$. Prove that there exists $\epsilon > 0$ such that $d(x, f(x))>\epsilon$, for ...
0
votes
2answers
21 views

Is a closure a disjoint union of limit points and isolated points

Definition) A point $x\in X$ is a limit point of S if every ball $B(x;r)$ contains infinitely many points from $S$. A point $x\in X$ is called an isolated point of S if $\exists r > 0$ such that ...
2
votes
2answers
56 views

A topological space which is Frechet but not Strictly-Frechet.

Let $X$ be a topological space and $q \in X$. $X$ is strictly Frechet at $q$, if, for all $A_n \subset X, q \in \bigcap_{n \in \omega} \overline {A_n}$ implies the existence of a sequence $q_n \in ...
0
votes
2answers
23 views

Is sequence convergent in subspace of compact metric space?

Problem is as follow. Let X be a compact metric space and A be a closed subset of X. Prove that every sequence in A has a convergent (note: convergent in A) subsequence. It is from my note. My ...
0
votes
1answer
18 views

difference between uniform topology and product topology

can anyone make an example that implies the difference between uniform topology and product topology on $\Bbb{R}^\infty$ ($\Bbb{R}\times \Bbb{R} \times \Bbb{R} \times \Bbb{R} \times ....$)?
6
votes
0answers
36 views

Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
1
vote
1answer
14 views

Generalisation of Tietze Extension Theorem for Compact Hausdorff Spaces

Let $X$ be a normal space and $A$ a closed subspace of $X$. Let $Y$ be a compact Hausdorff space. Is there a theorem that allows any continuous $f : A \rightarrow Y$ to be extended to a continuous $F ...
0
votes
1answer
35 views

Question about degrees of maps from $S^1 \rightarrow S^1$

Note: Since the degree of a map is independent of the base-point I'll speak loosely and just say $\pi_1(S^1)$. One definition of the degree of a map $f$ from the circle to itself is the number $k$ ...
7
votes
3answers
135 views

Topology - Product space example

In the book "Topology" from Boto von Querenburg I read the following example for product spaces: "The product space of a circumference and an interval $[a,b]$ with $0<a<b$ is homeomorphic to ...
1
vote
1answer
28 views

Show that if $h,k: S^1\rightarrow S^1$ are homotopic, they have the same degree.

We define the degree of a continuous map $h: S^1 \rightarrow S^1$ as follows: Let $b_0$ be the point $(1,0)$ of $S^1$; choose a generator $\gamma$ for the infinite cyclic group $\pi_1(S^1,b_0)$. If ...
0
votes
1answer
31 views

Maps from $SO(3)$ to $S_1, S_2$, and $S_1 \times S_2$

I am looking for continuous maps between the special orthogonal group of 3x3 matrices and the unit circle, unit sphere, and their product (S1, S2, S2 x S3, respectively). Any hints as to what I should ...
3
votes
4answers
486 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
0
votes
2answers
40 views

Suppose $X,\tau$ and $Y,\tau '$ are contractible to $x_0$ and $y_0$ respectively. Prove that $X\times Y$ is contractible and simply connected.

Here is what I got so far. Assume that $X,\tau$ and $Y,\tau '$ are contractible to $x_0$ and $y_0$ respectively. Since $X,\tau$ is contractible to $x_0$ $$id_X \sim f(x)=x_0 $$ meaning there ...
8
votes
2answers
31 views

Product topology and standard euclidean topology over $\mathbb{R}^n$ are equivalent

I would like to know why the product topology and the standard euclidean topology over $\mathbb{R}^n$ are equivalent. I already found the question here: Showing that the product and metric topology ...
0
votes
0answers
48 views

Does maximal principle imply open mapping theorem for any continuous function?

At first I spent a lot of time looking for counterexamples because I had never seen such a claim that M.P. implies O.M.T.. But later I realized the claim might be true, so I just had a try and proved ...
1
vote
2answers
90 views

Let $p: E\to B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite, then $E$ is compact. [duplicate]

So I start off and assume that some $\{U_\alpha\}$ is a cover of $E$. I want to reduce this cover to a finite subcover of $E$. Since $p$ is a covering map it is also an open map, therefore ...
1
vote
0answers
36 views

topological equivalence on interior of $D^2$ that is not continously extendable to $D^2$

As said in the title, I'm trying to find a topological equivalence on the interior of $D^2$ that is not continously extendable to $D^2$. I have an idea about this, so here it goes: Let ...
3
votes
0answers
26 views

Smash products of pointed spaces is really not associative

The canonical bijective map $\mathbb{N} \wedge (\mathbb{Q} \wedge \mathbb{Q}) \to (\mathbb{N} \wedge \mathbb{Q}) \wedge \mathbb{Q}$ is not an isomorphism of pointed spaces (i.e. homeomorphism), see ...
4
votes
2answers
35 views

For $f$ a continuous topological mapping, when are the values on the boundary of a set determined?

Suppose $f:X\to Y$ is a continuous map between topological spaces, and suppose we know the value of $f$ on a subset $S\subset X$. Continuity tells us that $f(\bar{S})\subset \overline{f(S)}$ for any ...