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; ...

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1
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3answers
22 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
0
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2answers
29 views

Showing that stereographic projection is a homeomorphism

For any $n\geq 0$,the unit $n$-sphere is the space $S^{n}\subset \mathbb{R^{n+1}}$ defined by $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
0
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1answer
25 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
0
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1answer
16 views

Precompactness vs. Separability

I was always wondering... Given a metric space. To what extend do these notions differ: $$\Omega\text{ precompact}\implies\Omega\text{ separable}$$ (Precompactness meaning totally bounded.)
6
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1answer
28 views

Books or texts on singularity theory

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
0
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0answers
15 views

>If $A$ is an uncountable subset of a space whose topology has a countable base, then some point of $A$ is an accumulation point of $A$. [duplicate]

If $A$ is an uncountable subset of a space whose topology has a countable base, then some point of $A$ is an accumulation point of $A$. I've shown that there is an injection from A to the power ...
2
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1answer
35 views

Strange Quotient space $X / \mathbb{Z}$

For a practice-exam exercise I am trying to understand why $X/ \mathbb{Z}$ is homeomorphic to $S^1$. Here, $X = (-1,\infty)$, and $\mathbb{Z}$ is acting as an additive group on $X$ with the action: ...
3
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2answers
53 views

Series convergence and compact space

Let $K$ be a compact topological Hausdorff space. $\{x_n\}_1^\infty \subset K $ such that $x_i \not= x_j, i \not=j$ and $\{a_i\}_1^\infty \subset \mathbb{K}$. Show the folowing are equivalent: for ...
1
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0answers
34 views

Equivalence of sigma algebras on the set of probability measures.

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
3
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2answers
50 views

Finite groups and topological spaces

Can we connect topological spaces with groups as: For topological space $X$ take biective homomorfisms $\phi: X\to X$, then divide such homomorphisms on classes of equivalency $\phi_1 \equiv\phi_2$ ...
0
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1answer
40 views

Elementary Operations on Sets

Let $X$ be a set with subsets $A$ and $B$. Prove: a). $X \setminus (X \setminus A) =A$. $X \setminus A$ is the set of all points of $X$ which do not belong to $A$. Given $p \in X$, we will show that ...
1
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1answer
26 views

Fixed point of a map [on hold]

If g is a continuous map from U onto V in the complex plane, where U and V are homeomorphic to disks and U a proper subset of V. Must there be a fixed point? And if g is conformal, is this point ...
0
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0answers
6 views

Chain boundary (Topology)

Could anyone give me a topological definition of chain boundary, if possible one which could be integrated in further definitions (homology, quiver, bound chain and so on)??
3
votes
1answer
16 views

Open set in Hilbert Cube.

Any open set in the Hilbert Cube is the union of open subsets of the form $$U_1 \times ... \times U_n \times X_{n+1} \times .... \times X_{n+k} \times...$$ where $X_k := [0, \frac{1}{k}]$ for $k \in ...
3
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2answers
37 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
0
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1answer
36 views

Subsets of a topological space and isomorphisms $X\longrightarrow X$

Let $X$ a topological space, and fix a subset $U\subseteq X$. I would like to find a characterization of the class $ \qquad \qquad \qquad \qquad \quad \Omega_{U} = \{V\subseteq X\ |\ V\cong U \text{ ...
2
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0answers
36 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 ...
1
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1answer
32 views

Measurable Functions [on hold]

If $f$ is such that $\| f \|$ is measurable, does $f$ have to be measurable? Any help would be appreciated. Please proof your answer.
20
votes
10answers
879 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
30 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 ...
6
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2answers
44 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 ...
2
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1answer
39 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 ...
1
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0answers
22 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
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2answers
65 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, ...
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0answers
28 views

Is following matrix sets convex? [on hold]

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
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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
30 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
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0answers
33 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
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1answer
41 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
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0answers
28 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
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0answers
39 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
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0answers
27 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
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1answer
54 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 ...
2
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0answers
35 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
76 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
51 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
29 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
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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 ...
-3
votes
1answer
38 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
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2answers
56 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
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1answer
35 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
35 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
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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?
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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
52 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 ...