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)

5
votes
0answers
119 views

How to find a homeomorphism $\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ having certain properties

Let $n\ge 2$ and let $C$ be a cantor space in $\mathbb{R}^{n}$. That is, $C$ is homeomorphic to the cantor ternary set. Let $x$ and $y$ be two points in $\mathbb{R}^{n}-C$, and let $L_{xy}$ be the ...
1
vote
4answers
465 views

$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$

Question is to prove : $f : S^1 \to \mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$ I guess it would be helpful to use intermediate value theorem Assuming $f(x)\neq f(-x)$ then given ...
3
votes
1answer
65 views

An example of a $F_{\sigma\delta}$ subset of $[0,1]$ of measure $1$ which is not $F_\sigma$

I'm trying to understand Borel sets. I am looking for a visual (i.e., constructive) $F_{\sigma\delta}$ subset of $[0,1]$ of measure $1$ which is not $F_\sigma$. Any idea or suggestion would be ...
1
vote
1answer
157 views

If $A\subsetneq X, B\subsetneq Y$, X,Y are connected then $X\times Y- A\times B$ is connected [duplicate]

Question is to prove that : If $A\subsetneq X, B\subsetneq Y$, and $X,Y$ are connected then $X\times Y- A\times B$ is connected. I do not immediately see why this is true. So, I thought if i can ...
0
votes
1answer
26 views

Every $\kappa$-metrizable space has countable $o$-tightness.

The $o$-tightness of a space X is said to be countable (notation: $ot(X)\leq\omega$) if whenever a point a belongs to the closure of $⋃γ$ , where $γ$ is any family of open sets in $X$, then there ...
2
votes
2answers
41 views

sequence of bounded domain in $R^n$

Consider $$\Omega_1 \supset \Omega_2 \supset \cdots$$ a sequence of bounded, open and convex domains in $\mathbb R^n$, with all the inclusions strict. I want to prove that $\bigcap \Omega_k \subset ...
1
vote
1answer
127 views

Interchange $\inf$ and $\lim$?

Let $U \subseteq \mathbb{R}^n$ be an open set and $K \subseteq \mathbb{R}^n$ be a compact set. Suppose we have a continuous function $c : U \times K \rightarrow (0, \infty)$. Now define $$\rho(x) = ...
5
votes
1answer
117 views

Does the category of Simplicial Complexes have finite limits and colimits?

Does the category of Simplicial Complexes have finite limits and colimits? Does the geometric realization functor preserve them? Thanks!
3
votes
2answers
174 views

Generalization of the hairy ball theorem.

The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Can the hairy ball theorem be strengthened to say that there is no ...
3
votes
2answers
186 views

An Open Cover $\mathscr{F}$ of $2\mathbb{N}$ That Has No Finite Subcover

What is an open cover $\mathscr{F}$ for the set $2\mathbb{N}=\{2n:n\in\mathbb{N}\}$ that has no finite subcover? My initial answer is ...
0
votes
3answers
84 views

Find an example of non-locally finite collection

I got stuck on this problem and can't find any hint to solve this. Hope some one can help me. I really appreciate. Give an example of a collection of sets $A$ that is not locally finite, such ...
5
votes
2answers
262 views

Who proved that existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ was sufficient for HEP?

It is well known that the existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is necessary to make $\left(X,A\right)$ a pair having the homotopy ...
2
votes
2answers
57 views

$C_p(X)$ is a dense subspace of $\Bbb R^X$

The topology of $C_p(X)$ ($C(X,\Bbb R)$ with the topology of pointwise convergence) coincides with the topology induced in $C(X, \Bbb R)$ from the Tychonoff product $\Bbb R^X$, that is, $C_p(X)$ ...
1
vote
0answers
67 views

Miranda's book pag. 60

The author’s dealing with the connected sum of two topological spaces. He defines the map $\pi: X \sqcup Y \mapsto Z$ and endows $Z$ with the quotient topology. He also says that the natural ...
0
votes
1answer
42 views

Inequality for open set measures

How do you prove that, given a sequence of open sets $I_i$ (in $\mathbb{R}^n)$, it is possible, given $\varepsilon>0$, to find a s sequence of open sets $J_i$ such that $I_i \subset J_i$ (strict ...
2
votes
1answer
386 views

Any space $X$ with the indiscrete topology is compact.

Let $\tau_X = \{ \varnothing, X \} $. Let $A \subseteq X$, Let $O$ be an open cover of $A$. Since topology on $X$ is finite, then $O$ must finite too. Obviously, any subcover $O'$ of $O$ must be ...
0
votes
1answer
308 views

If $F_1$ and $F_2$ are disjoint closed sets then there exist disjoint open sets $G_1$ and $G_2$.

Use an Urysohn function to give a solution of this problem: Prove that if $F_1$ and $F_2$ are disjoint closed sets in $\mathbb{R}^n$, then there exist disjoint open sets $G_1$ and $G_2$ such that ...
0
votes
1answer
688 views

Open Cover Having No Finite Subcover

Information: Consider the open cover $\mathscr{F}=\begin{Bmatrix}\pmatrix{0,5-\frac{1}{n}}:n\in\mathbb{N}\end{Bmatrix}$ for the set $(0,5)$. Question: Is ...
2
votes
1answer
63 views

Subspace topology in the union of subsets

Let $A,B$ and $C$ be subsets of a topological space $X$ with $C \subseteq A \cup B$. If $A,B$ and $A \cup B$ are given the subspace topology, prove that $C$ is open with respect to $A \cup B$ if and ...
3
votes
2answers
244 views

Is there a first countable, 0-dimensonal, locally compact, lindelof, non-compact space?

Is there a first countable, 0-dimensonal, locally compact, lindelof, non-compact space in which all non-empty open sets have $\pi$-weight $\mathfrak c$? It also can be seen here. Thanks for your ...
1
vote
1answer
113 views

Characterization of totally bounded sets in Metric spaces

Searching material to know more about totally bounded sets, I found this property Let $(M,d)$ be a metric space. So $A\subset M$ is totally bounded if and only if for every $\varepsilon >0$ ...
3
votes
0answers
73 views

A connected sum and wild cells

Can we find such a connected sum of two spheres (in any dimension) that is not homeomorphic to the sphere? $\def\R{\mathbb R}$ It seems that there should be examples like that, because there are lots ...
3
votes
1answer
421 views

Non-T1 Space: Is the set of limit points closed?

I have shown that the set of limit points in $T_1$-space is closed, and the proof uses the $T_1$ axiom, so I was wondering: Given $X,$ not necessarily $T_1,$ and any $A\subset X,$ is it necessarily ...
10
votes
3answers
433 views

Statement about Homotopy in Brown's “Topology & Groupoids”

I am trying to understand a statement in Brown's Topology and Groupoids, 7.2.5 (Corollary 1), page 270. Let's first have some preliminary remarks Let $X,Y$ be topological spaces. The track groupoid ...
2
votes
4answers
155 views

Very basic intro to real analysis question

In page 2 of baby Rudin, Rudin wants to prove that $ \forall p \in \mathbb{Q} $ such that $ p>0 $ and $p^2 < 2$ there exists a $q \in \mathbb{Q}$ such that $q>p$ and $q^2<2$ In his ...
1
vote
1answer
56 views

Example of a Spread which is not Complete

This is a continuation of an original question about spreads, which are something like pre-branched covering spaces. See the basic definitions here: A Complete Spread I have an example of a spread ...
1
vote
0answers
78 views

understanding Poincare conjecture

I was reading a Wikipedia article on Poincare conjecture and found out that it basically says that if each loop in the space can be continuously tightened to a point then the space is topologically ...
0
votes
1answer
36 views

In a Baire space $X$, if an open set meets a nonmeager set, is the intersection nonmeager?

In a Baire space $X$, if an open set $U$ meets a nonmeager set $N$, is the intersection nonmeager? If they do not meet then it's false; take the upper half line of the reals. It does not meet some of ...
3
votes
0answers
175 views

What are some characterizations of the strong and total variation topologies on measures?

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for ...
2
votes
1answer
92 views

$X,Y$ equipped with discrete topology $\implies$ any $f:X \to Y$ is continuous

My attempt: Pick an open set $O \subseteq Y$ open in $Y$. So, $f^{-1}(O) \in X$. But since all elements of $X$ are open sets since $X$ enjoys the discrete topology, then $f^{-1}(O)$ must be open. ...
1
vote
1answer
90 views

Topology on the set of closed subsets of $\mathbb{R}^n$.

I have applications $\mathbb{R}\ni t\mapsto C(t)$ where $C(t)$ is a closed subsets of $\mathbb{R}^n$. I want to say that these applications are continuous/discontinuous. What topology can we put on ...
3
votes
2answers
95 views

Real analysis uniformly continuous

Is every differentiable function $f$ from the open interval $(0,1)$ to the closed interval $[0,1]$ uniformly continuous? No means please give counter example.
2
votes
1answer
309 views

Base and empty set of a topology

Given space X = {a, b, c}, $\beta$ is a basis for a topology $\tau$ on X. $\tau$ = { $\varnothing$, X, {a}, {b}, {a,b}}, $\beta$ = {{a}, {b}, X}. $\beta$ can't union its elements to get empty ...
1
vote
1answer
32 views

Is the intersection of a chain of covers of a set also a cover for that set?

Suppose $X$ be any non-empty set and $\mathcal{C} = \{\mathcal{U}_{\lambda}:\lambda \in \Lambda\}$ be a chain of covers of $X$. So for every $\lambda$, $X \subseteq ...
1
vote
0answers
293 views

Is my proof for the translation of an f-sigma set valid?

Let $F = \cup^{\infty}_{k=1} A_i$ where each $A_i$ is a closed set. Since $F$ is $F_\sigma$, every $F_\sigma$ set is measurable, and every measurable set is translation invariant, F is translation ...
13
votes
3answers
382 views

Metal Ball Cage Template Cardinality: A Brilliantly Lazy PROOF

N.B. - I'm looking for the simplest way to ascertain the number of templates $T$ (see below) comprising the structure from just one angle alone; that is, I'm sitting down looking up at this thing, ...
1
vote
2answers
88 views

a condition equivalent to compactness in linearly ordered spaces

Does anyone know where can I find a proof to this proposition: A linearly ordered topological space is compact if and only if every bounded subset has an infimum and a supremum. Thank you,
0
votes
1answer
50 views

A question about function on product spaces

Let $\{X_i:i\in I\}$ be a family of topological spaces. Is the function $f$ from $X=\prod\{X_i:i\in I\}$ to $X_K=\prod\{X_k:k\in K\}$, Where $K$ is a finite subset of $I$, continuous and open? The ...
3
votes
1answer
115 views

Is this topology discrete?

If a topology $\tau$ is strictly finer than cocountable topology and lower limit topology, then $\tau$ is discrete topology on $\mathbb{R}$. Prove or disprove.
8
votes
1answer
526 views

Is a connected sum of manifolds uniquely defined?

It is a standard excercise in differential geometry to prove that a connected sum $M\#N$ of two smooth manifolds $M,N$ of the same dimension is uniquely defined (under some assumptions regarding ...
0
votes
2answers
42 views

Testing for Topology

Is $\tau=\lbrace G: G\subset \mathbb{Q}$ or $\mathbb{Q}\subset G\rbrace$ is a topology on $\mathbb{R}$? I think this is not. If we choose $G_n=\left\lbrace ...
3
votes
1answer
58 views

Is the finite topology of an endomorphism ring Noetherian if the elements of its base satisfy the ACC

Given a left R-module M over some ring R, the finite topology on its endomorphism ring E=End(M) is defined by defining the neighborhoods of zero as the sets $$U(x_1,\ldots , x_n) = \{ f\in E \mid ...
6
votes
1answer
123 views

Fundamental group of real $3\times 3$ matrices with rank $1$

In "Manetti - Topologia" there is the following exercise: Compute the fundamental group of real $3\times 3$ matrices with rank $1$. He suggests to show that there is a covering map of degree ...
2
votes
2answers
53 views

Whether these induced topologies are comparable?

$$\|x\|_1=\sum_{i=1}^{n} |x_i|$$ and $$\|x\|_2=(\sum_{i=1}^{n}|x_i|^2)^{1\over 2}$$ these two norm induce topologies on $\mathbb{R}^n$, I want to know whether they are comparable?
1
vote
1answer
92 views

Question about Product Topology

I know if X = {a,b} and Y = {c,d,e}, then X $\times$ Y = {(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)} However,I am confusing when seeing the follow example: Take the topology $\tau$ = {$\varnothing$, ...
1
vote
1answer
390 views

Boundary of compact set

My question is: Is the boundary, $\partial K=\overline{K}\setminus \operatorname{int}(K)$, of any subset $K$ of any topological space $X$ necessarily compact?
2
votes
2answers
819 views

Nested sequence of closed sets

Is it true that every nested sequence of non-empty closed sets $(I_n)$ (one such that $I_{n+1} \subset I_n$) has a non-empty intersection?
1
vote
1answer
530 views

compact and countably compact

As we know that a topological space $( X, \tau)$ is said compact if every open cover for $X$ has a finite subcover, and a topological space $( X, \tau)$ is said countably compact if every countably ...
4
votes
3answers
346 views

$\mathbb R$ is not isometric with $\mathbb R^2$

show that $\mathbb R$ is not isometric with $\mathbb R^2$ (with the usual metrics). I want to use the first definition of continuity (i.e. the $\epsilon $ -$\delta$ stuff) but I don't see a way to ...
0
votes
1answer
99 views

What is 'target manifold'?

I saw in a lecture about O(3) sigma model something about 'target manifold', but I do not know what is it. Is there any book I could learn about that?