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 (3)

0
votes
0answers
5 views

Question about metric spaces(compact, dense) [duplicate]

Prove, that every compact metric space has a countable, and dense sub-set. I don't know how I should prove this, I tried with the definition: A topological space X is called compact if each of its ...
0
votes
0answers
7 views

$\sigma-$algebra generated by $I_A$ is same as by $I_{f}$

Let $E$ be a local compact, polish room (if you need it you can set $E = \mathbb{R}^n$) Let $(M(E)$ be the room of all radeon measures on $E$ $$I_f: M(E) \to \mathbb{R}, \mu \mapsto \int_E f d\mu, f ...
0
votes
1answer
7 views

Union of family of topology is a subbasis

For any family $(\tau_\alpha)_{\alpha \in A}$ of topologies on a set X, How do we show the fact that $\cup_{\alpha\in A} \tau_\alpha$ is a subbasis for a topology on X that is stronger than all the ...
0
votes
0answers
12 views

$X$ is Frechet Compact iff $X$ is compact.

I have done the proof that $1)\ X$ is Frechet Compact iff $X$ is sequentially compact. $2) \ X$ is sequentially compact iff $X$ is compact. Thus we can conclude that $X$ is Frechet Compact iff ...
2
votes
1answer
29 views

Wot convergence and sot convergence

Let $\{A_n\} $ be a sequence of bounded linear operators on Hilbert space $H$ and $\langle A_n\xi,\eta \rangle \to \langle A \xi,\eta\rangle$ for $\xi,\eta\in H$ with $\|\eta\|=1$. Show that $\|A_n\xi ...
0
votes
1answer
16 views

Difference of open sets

Let $A$ and $B$ be two open sets of $\mathbf{R}^2$ (with the Euclidean topology). Is it true that if $A\setminus B$ is non-empty then there exists an open ball contained in $A\setminus B$?
3
votes
1answer
38 views

Proof review- Every sequence in $\mathbb{R}$ has monotone subsequence

I would like to know if my proof is correct. I'm worried that I may have broken some rules of constructive proofs (e.g. providing a construction with infinite steps). Also, please excuse my abuse of ...
0
votes
1answer
29 views

An irrational flow on a torus is dense

I was surprised I couldn't find the proof of this here. The problem is to prove the image of $\{(r,r\sqrt 2)\mid r\in\mathbb R\}$ is dense in the torus where we think of the torus as $I\times I$ with ...
0
votes
1answer
19 views

Continuity of Functions with Sets Equipped with Subspace Topology

Let $A, B$ be closed subsets with the subspace topology in the topological space $X = A \cup B$. Let $g:A \to Y$ and $h:B \to Y$ be continuous. Prove that if $g(x) = h(x)$ for all $x \in A \cap B$, ...
2
votes
1answer
27 views

What is a parametrized surface? How is it different from a surface? (Multivariable Calculus)

My textbook defines it like this: Let F be a continuous function from a subset D(F) R2 into Rq. Suppose that D(F) is pathwise connected, and that every point in D(F) is either an interior point of ...
3
votes
4answers
64 views

Why dont we say if $f(U)$ is open for every open set $U$ in $A$, then $f$ is countinous?

Definition of continuity in topology; Let $f :A \to B$. If $f^{-1}(U)$ is open in A for every open set U in B, we say f is countinous. Why dont we say if $f(U)$ is open for every open set $U$ in ...
0
votes
1answer
27 views

Non-equivalence of norms.

What ways to prove the non equivalence of two norms? Is it sufficient to find a sequence associated to the two norms and having two different limits to prove the non equivalence?
0
votes
1answer
25 views

On embedding a sort of $CW$ complexes to a Euclidean space.

I'd like to know if a finite dimensional, locally finite, $CW$ complex with countable cells can always be embedded to a Euclidean space. All I know is that it holds in the case $\dim=1$.
2
votes
0answers
34 views

Topology and Planetary Nebulae

I apologize ahead of time if this receives any down-votes, but I was just reading a text on topology when the idea struck me: has any mathematician or, for that matter, any topologist, analyzed the ...
0
votes
0answers
31 views

Gromov compactness theorem

Reference: this book, page 493. For a compact metric space $X$ define $\text{Cov}(X,\epsilon)= \min \{n \, : \, X \text{ is covered by $n$ closed } \epsilon\text{-balls} \}$ and ...
0
votes
1answer
42 views

System of equations and the Brouwer's Fixed-Point Theorem.

Let's consider the following system of equations: \begin{eqnarray}{ e^{xyz} = \frac{x}{\sqrt{e^{2xyz}+1}}\\ \cos(x+y+z) = \frac{y}{\sqrt{e^{2xyz}+1}}\\ \sin(x+y+z) = \frac{z}{\sqrt{e^{2xyz}+1}} ...
2
votes
1answer
31 views

Example of converging subnet, when there is no converging subsequence

I'm trying to wrap my head around the concept of nets/subnets, especially in the following example. Let $X$ be the Banach space $\ell_{\infty}$ and $X^*$ its dual. We know by Banach-Alaoglu that the ...
0
votes
1answer
34 views

Showing properties of Cantor space

If we have the proposition that Any subset of the real line that is compact, totally disconnected and perfect is homeomorphic to the Cantor Set. And consider the Cantor space $2^{\omega}= ...
4
votes
0answers
52 views

Finding a good cover such that its lifting is still a good cover

Let $Y$ be a compact manifold, $X$ a topological space and $f: X \to Y$ a surjective map. Suppose further that every point in $Y$ has arbitrarily small open neighbourhoods such that their preimages in ...
-3
votes
0answers
30 views

Countable topological spase have a countable base? [on hold]

Prove or give a counter-example. Every countable topological space X have a countable base.
1
vote
1answer
36 views

Continuity and interior

I have questions about the relation between continuity and interior based on the article ;Continuity and Closure At first I guess that there will be a property like $f:X\rightarrow Y$ is continuous ...
2
votes
1answer
44 views

Theorem 20.4 in Munkres' TOPOLOGY, 2nd edition: How are these three topologies different on an infinite Cartesian product of $\mathbb{R}$ with itself?

The standard topology on $\mathbb{R}$, the set of real numbers, has as a basis all open intervals $(a,b)$, where $a$, $b$ are real numbers such that $a < b$. Let $J$ be an arbitrary (finite, ...
1
vote
3answers
61 views

Is the perimeter of a square isometric to the circle? Homeomorphic?

An isometry is defined as a function that preserves distances between points two metric spaces. If make a circle with the same perimeter/circumference as the square, is it possible to find an ...
1
vote
4answers
30 views

Showing there is a bijection from all open subsets to all closed subsets of $M$

(From Pugh's RMA) Let $\mathcal{T}$ be the collection of open subsets of a metric space $M$, and $\mathcal{K}$ be the collection of all closed subsets. Show there is a bijection from $\mathcal{T}$ to ...
6
votes
3answers
74 views

Showing that arcs do not separate the plane $\mathbb{R}^2$

Is $\mathbb{R}^2\setminus f([0,1])$ connected if $f:[0,1]\to\mathbb{R}^2$ is an embedding? It seems that this is clearly true but I am having a hard time proving it. The only things that I know is ...
1
vote
0answers
64 views

Manifolds and CW-complexes

This is a very naive question. Every manifold (assumed to be paracompact) is a CW-complex? Thanks.
2
votes
2answers
50 views

Totally disconnected implies nowhere dense

Why is it true that a totally disconnected space implies it is nowhere dense in the reals? I know that totally disconnected implies the component is a singleton, but how do we construct a nowhere ...
2
votes
0answers
49 views

A proposition of Urysohn's Lemma in real analysis

I have a problem proving the following theorem, which appears in Evans' PDE text book: Let $K$ be a compact set in space $R^n$, and $K\subseteq \cup_{i=1}^{k} U_{i},k\geq2$, where $U_i$ are open ...
1
vote
0answers
30 views

Connection between Chladni Plates and Algebraic Topology?

Does anybody know of a connection between Chladni Plates and Algebraic Topology? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
2
votes
0answers
38 views

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image? [duplicate]

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image ? that is if $X,Y$ are Hausdorff spaces and $f:X \to Y$ is continuous such that for any ...
0
votes
2answers
21 views

Questions on open set Cantor Space

I am trying to show that each point in the Cantor space, $\prod_{i \in\mathbb{N}}\{0,1\}_i$ is a limit point. But I am confused about what is the open set of the Cantor space. Thanks!
1
vote
1answer
43 views

To prove Heine-Borel theorem for $\mathbb R^n$ with usual Euclidean topology

To prove that any closed and bounded subset of $\mathbb R^n$ is compact , I proceed as : Since $\mathbb R^n$ is complete so any closed subset of it is complete . Then I show that any bounded subset of ...
1
vote
0answers
35 views

bounded continuous function are open in set of continuous functions?

Let $D$ be a metric space and $K \subset D$ a compact set. Let $C_b(K) = \{ f \in C(K, \mathbb{R}) | \mbox{ f is continuous and bounded } \}$. I want to prove that $C_b(K)$ and B(K,1) := $\{ f \in ...
4
votes
1answer
53 views

Homeomorphisms in $\mathbb{R}^2$.

I'm getting confused about homeomorphisms. I believe that $[0,1]\times [0,1)$ and $[0,1)\times [0,1)$ are homeomorphic but $[0,1]\times (0,1)$ and $[0,1)\times (0,1)$ are not. Please can you try and ...
4
votes
2answers
79 views

Showing that two spaces are homeomorphic

I was trying to show that a torus is homeomorphic to $S^1 \times S^1$ , I tried to work with the fundamental group of both, which are equal, but that doesn't imply they're homeomorphic, (at least i ...
1
vote
1answer
24 views

Is there a topological space $X$ such that $C_p(X)$ is sequential but not Fréchet?

Is there a topological space $X$ such that $C_p(X)$ is sequential but not Fréchet? Definitions: A topological space $X$ is called sequential if whenever $A \subset X$ contains all limits of ...
0
votes
1answer
12 views

Questions about total disconnectedness

I have come across the proof that the Cartesian product of totally disconnected sets is also disconnected in the following post. Product of totally disconnected space is totally disconnected? ...
11
votes
2answers
86 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}^2$ is closed

Are there a closed subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? I cannot think about a continuous and bijective function ...
5
votes
2answers
41 views

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$.

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$. We assume on the contrary that there does not exist ...
3
votes
2answers
76 views

Find all compact sets in $\mathbb{R}$

In $\mathbb{R}$, considering the topology consisting of the empty set and all sets containing $0$ and $1$, I need to find all compact sets. I understand the definition of a compact set but don't know ...
-5
votes
1answer
39 views

Examples of a infinite dimensional simplicial. [on hold]

I want see an example of infinite dimensional simplicial, diferent to the examples built using ergodic theory.
1
vote
0answers
12 views

Basis and Subbasis verification

Suppose I have a set X = {a,b} and the topology T = {{a,b},{a},{b},{0}} where 0 is the empty set. Then a basis for the topology T is {{a,b},{b}, {a}} and also the subbasis is {{a},{b}}. Is any of this ...
3
votes
1answer
29 views

Is the Jordan Curve Theorem True for Curves on a Toroid

I'm making a presentation on the Jordan Curve Theorem, and I want a counter-example. I think that the toroid is a great example, but I want to make sure that I'm correct. Since the toroid has a ...
-1
votes
2answers
38 views

Let $X$ be a topological space. Prove that for any $x $ in the intersection of all opens sets $=\{x \}$, the space $X$ need not be Hausdorff.

Let$ X$ be a topological space. Prove that for any $x$ in the intersection of opens sets $=\{x \}$, the space $X$ need not be Hausdorff. My thoughts / strategies. I want to choose some other ...
0
votes
0answers
24 views

gluing together continuous function 2

The problem that I was working is exactly this and the approach given by Ronnie Brown is simple and beautiful. But as was noted in the other post by Henning Makholm the proposition as stated is not ...
1
vote
0answers
24 views

Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...
0
votes
1answer
32 views

gluing together continuous functions

HI I was checking and old question here and I have troubles to proof the following: Proposition: Let $X$ be a space with subspaces $Y,A,B$ such that $X \backslash Y= A \sqcup B$ (disjoint union). Let ...
0
votes
0answers
29 views

Does there exist an open subset U of $R^{n+1}$ and a bijective function $f: S^n→U$? [on hold]

Let $S^n=\left \{ x_1,x_2,...,x_{n+1}| x^2_1+x^2_2+...+x_n^2=1 \right \}$ Does there exist an open subset U of $R^{n+1}$(natural topology) and a bijective function $f: S^n→U$ ?
1
vote
1answer
70 views

$f(X)$ is uncountable and hence $X$ is uncountable.

My question: let $f : X \to \Bbb R$ be a non-constant continuous function on a connected metric space and assume that $f(X)$ is uncountable; then $X$ is uncountable. We know continuous image of a ...
3
votes
2answers
34 views

If $S \subset T$ prove $\overline{S} \subset \overline{T}$ and $\text{int}(S) \subset \text{int}(T)$

Closure: $\bar{S} = \cap K$ where $K$ ranges over all closeds containing $S$. Interior: $int(S) = \cup U$ where $U$ ranges over all opens contained in $S$. My attempt for first part: Let $x$ be an ...