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)

1
vote
0answers
21 views

Set of non fixed points of an automorphism

I am trying to prove the following "For an orbifold chart $ (\tilde{U},G,\phi)$ the set of non fixed point of $ g : \tilde{U} \rightarrow \tilde{U} $ where $ 1 \neq g \ \in G$ is dense in $\tilde ...
0
votes
1answer
23 views

Combinations Of Measurable Functions [on hold]

If $f$ is such that $\| f \|$ is measurable, does $f$ have to be measurable? Any help would be appreciated. Please prove your answer.
10
votes
7answers
212 views

Surprising applications of topology [on hold]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
0
votes
1answer
21 views

A bijection from a disconnected space to a connected space?

Can we find an bijective continuous map $f:X\to Y$ from a disconnected topological space $X$ to a connected topological space $Y$? It seems counter intuitive for me, but I am not able to prove that ...
5
votes
1answer
29 views

Counter example to Mostow's rigidity theorem for 2-manifolds.

I am trying to understand a counter-example to Mostow's rigidity theorem. Here is the counter example I want to understand. Take two non-isometric octagons with the sum of interior angles equal to ...
1
vote
0answers
22 views

Is a dense and co-dense subset $G_\delta$ or co-$G_\delta$

Let $A \subset \mathbb{R}$ such that $A$ and $A^C$ are both dense. By Baire's Theorem at most one of $A$ and $A^C$ is $G_\delta$ (i.e. a countable intersection of open sets) I couldn't think of an ...
0
votes
0answers
12 views

On a congruence for the number of finite topologies

I am making search about "On a congruence for the number of finite topologies". I have found a paper. I guess it is written in Russian. How can I find English version of this paper ? I am also ...
0
votes
2answers
48 views

Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
0
votes
0answers
25 views

Is following matrix sets convex?

Given $A\in\{0,1\}^{n\times n}$. Denote $\mathcal{A_{n,n}}$ to be collection of rank $1$ matrices from $\{0,1\}^{n\times n}$. Denote $\mathcal{A_{n,n}}[A,c,S\subseteq\Bbb R,T\subseteq\Bbb ...
1
vote
2answers
44 views

for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed?

for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed in the metric space $(\mathbb{Q},d)$ where $d(x,y) = |x-y|$ my attempt: I suspected it's closed for all real numbers: let $x,y \in ...
2
votes
1answer
26 views

In a locally compact Hausdorff space, why are open subsets locally compact?

Let $X$ be a locally compact Hausdorff space, and $A \subset X$ closed. I want to show that $X - A$ is locally compact. I have found a proof here: Open subspaces of locally compact Hausdorff spaces ...
0
votes
0answers
24 views

The triangle inequality for shortest paths of graphs

In why-the-triangle-inequality I found the statement: for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite ...
2
votes
0answers
24 views

Extending a continuous map between the boundary of two cells.

I'm working in Lee's book on topological manifolds and have gotten stumped on the first question in chapter 5, the chapter on cell complexes. The problem is: Let $D$ and $D'$ be two closed cells ...
0
votes
0answers
25 views

“Absolute retracts” in arbitrary category

Is there a standard notion of something like "absolute retract" in arbitrary categories that generalizes absolute retracts in topology? I am mostly interested in categorical approach to Hausdorff ...
3
votes
0answers
35 views

Is there a way to define the concept of manifolds so it looks more like “generalised affine spaces”?

What I have in mind is along the lines of this: Let $M$ a topological space, $V$ a normed vector space, and $$ \boxminus \colon M\times M \to V, $$ $$ \boxplus \colon M\times V \to M. $$ Then ...
1
vote
0answers
25 views

how to conclude a subset of $M_n(\mathbb{C})$ is compact from spectral radius?

could any one tell me which of the following is/are compact subset? $S=\{A\in M_n(\mathbb{C}): \rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): A=A^*,\rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): ...
2
votes
1answer
53 views

set of all $2\times 2$ matrcies having neither eigen value is real

Could any one tell me whether the following subsets of $M_2(\mathbb{R})$ are open, closed or neither open nor closed? set of all $2\times 2$ matrcies having neither eigen value is real. set of all ...
1
vote
0answers
30 views

What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
2
votes
2answers
33 views

Direct sum of metrizable spaces.

I managed to prove that an arbitrary direct sum of metrizable spaces is again metrizable. However, I used the theorem that says that a hausdorff regular space is metrizable if and only if there existd ...
6
votes
3answers
73 views

Exposed point of a compact convex set

I'm trying to show that given a compact convex set $K$ in $R^d$, there must be at least one exposed point (where $v$ is exposed if there exists a hyperplane H such that $H \cap K = \{v\}$ . This is a ...
4
votes
3answers
49 views

1-1 correspondence between [0,1] and [0,1) [duplicate]

I wonder how to build a 1-1 correspondence between [0,1] and [0,1). My professor offers an example such that 1 in the first set corresponds to 1/2 in the second set, and 1/2 in the first set ...
-1
votes
2answers
25 views

prove of topology and metric spaces [on hold]

Prove or disprove $f: A \to B$ a function from $A$ to $B$. $A_i$ subset of $A$ and $B_i$ subset of $B$. If $A_0 \subset A_1$ then $f(A_0) \subset f(A_1)$ $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$ ...
1
vote
1answer
13 views

Strong Topology and Strong Operator Topology on Hilbert Space

Suppose $H$ is a Hilbert space (much of this still works if it's just a Banach space), $x\in H$, and $(x_n)$ a sequence in $H$. Does $x_n\to x$ strongly in H iff $x_n\to x$ as operators in the strong ...
-2
votes
1answer
37 views

Question about a topology proof [on hold]

Hi. I need help with this simple question. I am not able to get this one.
3
votes
1answer
47 views

Small exercise in topology

I have a small question i have a topological space $(\mathbb{N},\tau)$ where $\tau=\{\emptyset,,\mathbb{N},\mathbb{N}^*, \{A_n\}_{n\in\mathbb{N^*}}\}$, $A_n=\{1,2,....,n\}$ and we consider the set ...
3
votes
2answers
54 views

Constructing A Space Filling Curve that fills the Unit Square

I'm reading Neal Carothers' Real Analysis, and he constructs a curve defined over $[0,1]$ that fills the unit square as follows: Let $f$ be a real-valued function over $[0,1]$ such that $f$ is $0$ ...
2
votes
1answer
34 views

Polish spaces, closed sets and $G_{\delta}$ sets

In a series of lecture notes regarding descriptive set theory, in the section regarding the Borel hierarchy I found the following statement: We will restrict ourselves from now on to Polish ...
2
votes
0answers
28 views

Following problem on topology $(N.B.H.M - 2015)$

let $X = \{ f \in C[-5 , 5] : f(-5) = f(5) = 0 \}$ . Then Which of the following statement are true : (a) There exist $f \in X$ such that $f \equiv 2$ on $[-1 ,0 ]$ and $f \equiv 3$ on ...
1
vote
2answers
41 views

Let $a$ and $b$ be arbitrary real numbers with $a < b$. Show that $[a,b]$ is closed by proving its complement is open.

Let $a$ and $b$ be arbitrary real numbers with $a < b$. Show that $[a,b]$ is closed by proving its complement is open. I don't have any idea on this, can anyone help me on this?
-2
votes
0answers
47 views

Help with general topology questions [on hold]

Given $P_0=(x_0,y_0)$ and $P_1=(x_1,y_1)$ points in $\mathbb{R}^2$, define the distance between $P_0$ and $P_1$ as $$d(P_0,P_1)=\sqrt{(x_0-x_1)^2+(y_0-y_1)^2}.$$ In $\mathbb{R}^2$, the equivalent of ...
-3
votes
3answers
51 views

Proof of questions with general topology. [on hold]

Let $A$ be any subset of $\Bbb R$ with $|A| < \infty$. Prove that $A$ is closed. Can anyone please help me with this proof?
-2
votes
3answers
40 views

doubt with proof in genral topology [on hold]

let Z and Q represent the integers and the rationals, respectively. prove that Z is a closed subset of R. Frankly I don't have an idea how to start. Can anyone please help me with this proof.
1
vote
1answer
33 views

How does look like an open set in one point compactification?

How does look like an open set in one point compactification? $X$ is that space and $Y$ is its one point compactification. Is it: $U$ open in $Y$ if it is open in $X$ or if $U=Y\backslash C$, for ...
1
vote
1answer
46 views

Prove that, for $x \in \mathbb R$ and $\delta_x > 0$, the open interval $(x-\delta_x, x+\delta_x)$ is itself an open set [duplicate]

Prove that, for $x \in \mathbb R$ and $\delta_x > 0$, the open interval $(x-\delta_x, x+\delta_x)$ is itself an open set. I am preparing for my exam and we will be asked to prove various ...
-4
votes
1answer
33 views

doubt with topology and functional analysis [on hold]

Prove that if $x \in \mathbb R$ and $\delta(x) > 0$ in the interval $(x-\delta(x), x+ \delta(x))$ is itself an open set. How to prove this can anyone help me on this?
-1
votes
1answer
26 views

Is every point of rational number boundary point?

While studying first chapter of multivariable calculus, I am wondering if every point of the rational number is boundary point. It is obvious that $\Bbb{R}^n$ is the union of interior, exterior, ...
2
votes
0answers
52 views

Is a “network topology'” a topological space?

Is there any connection between the computer science phrase "network topology" and the mathematical notion of a topological space (or, is there any other way to connect "network topologies" with ...
1
vote
1answer
22 views

If $p:A\to B$ and $q:C\to D$ are quotient maps, $B$ and $C$ locally compact, separable spaces, is $p\times q$ a quotient map?

It is a true or false question from an old test. At first I tried some counterexamples, using the line with two origins or taking $B$ as a quotient space of the real line by some not-open subset, ...
2
votes
1answer
29 views

If a continuous function is nonzero at a point $a$, there is a ball around $a$ in which it has the same sign as $f(a)$

Let $f$ be a scalar field continuous at an interior point a of a set $S\in \mathbb{R}$. If $f(a)\ne 0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The ...
1
vote
0answers
18 views

Entropy of isometric extension

A similar question to mine was asked before at the address below but it was not answered there so I am asking it again. Also there is a more specific case I am interested in. Topological entropy of ...
2
votes
1answer
33 views

Visualisation of Compact Metric Spaces

How can I visualise a compact metric space? It is a space of which every infinite open cover has a finite subcover. If I try to imagine finitely many open balls covering a space wholly, it seems to ...
0
votes
0answers
11 views

Continuous scalar field at an interior point of S and same sign proof.

Let $f$ be a scalar field continuous at an interior point $a$ of a set $S \in R$. If $f(a)$ is not $0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The above ...
2
votes
2answers
44 views

How to prove the closure $\bigcup_{n\ge 1} F_{n}$ is totally bounded and closed

Let $(X,p)$ be a metric space. Write $F$ for the set of subsets of $X$ which are closed, bounded, and non-empty. For each integer $n\geq1$, write $F_{n}$ for the set of subsets of X which are finite ...
1
vote
0answers
19 views

Example of subset of $\mathbb{R}²$ such that $A\neq A'\neq A''\neq A'''$? [duplicate]

I am looking for a subset of $\mathbb{R}²$ such that $A\neq A'\neq A''\neq A'''$ (where $A'$ is the set of limit points of $A$). I read it's possible but I don't even see how it could be ... I've ...
1
vote
0answers
28 views

Topological spaces from compact Hausdorff zero dimensional spaces

I saw a construction of general topological spaces using compact Hausdorff zero dimensional topological spaces, but I have no clue now of the construction or reference to this. I would be thankful if ...
2
votes
1answer
29 views

variations of Kuratowski closure complement theorem

I have been reading about the Kuratowski closure-complement theorem from the paper "THE KURATOWSKI CLOSURE-COMPLEMENT THEOREM by B.J. Gardner and M. Jackson'. It states that: If $(X,\tau)$ is a ...
-2
votes
1answer
24 views

How can I prove that it isn't a compact space [on hold]

Let $X=N$ and $B$ is a base for topology $τ(B)$ on $N$ . $B$={φ,{0,1,2,3},{4,5,6,7},{8,9,10,11},........} how can I prove that ($N$,$τ(B)$) is not compact space
2
votes
1answer
18 views

Closure of intersection with vector subspace

I am confused with the footnote on page 198 of http://www.ma.huji.ac.il/~razk/iWeb/My_Site/Teaching_files/TVS.pdf Essentially: Let $X$ be a topological vector space and $Y$ a finite-dimensional ...
1
vote
2answers
41 views

Question about Rudin's example of topological space

I began reading Rudin's Real and Complex Analysis, and I have a question about the following: Rudin defines a topology $T$ in a set $X$ as the collection of subsets of $X$ such that (i) empty set ...
1
vote
2answers
52 views

Prove or disprove that this function is continuous

If $f(x,y)$ is a real valued continuous function defined in $A \times B$ where $A$, $B$ are compact sets in $\mathbb R^n$ and $\mathbb R^m$ respectively. Let $g(x)=\min_{y \in B}f(x,y)$. Prove or ...