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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4 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 ...
2
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28 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 ...
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0answers
18 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
29 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 ...
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18 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
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2answers
29 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
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3answers
66 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
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3answers
47 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 ...
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2answers
24 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 $f(A_0) \subset f(A_1)$ $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$ ...
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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 ...
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1answer
36 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
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1answer
44 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
53 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
33 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
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0answers
25 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 ...
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2answers
39 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
45 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
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3answers
49 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?
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3answers
39 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.
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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 ...
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1answer
45 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 ...
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1answer
31 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?
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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
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0answers
50 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
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1answer
19 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
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1answer
28 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 ...
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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
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1answer
32 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 ...
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10 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
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2answers
43 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 ...
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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 ...
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0answers
27 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
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1answer
24 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 ...
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1answer
23 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
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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 ...
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2answers
40 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 ...
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2answers
51 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 ...
2
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1answer
30 views

Why not $(a,b)$ is not possible to define $\rho(f, g)$?

According to C.Adam's Topology: I don't know about compactness, but before introducing compactness in this book, in one of exercises it is asked: "Explain why we cannot generally define $\rho(f, g)$ ...
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2answers
44 views

Show that $d_V$ is a metric

Problem: For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ in $\mathbb{R}^2$ define: $d_V(p,q) = \begin{cases}1 & p_1\neq q_1 \ or\ |p_2 - q_2|\geq 1 \\ |p_2 - q_2| & p_1= q_1 \ and\ |p_2 ...
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0answers
27 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
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35 views

Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
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1answer
21 views

Suppose a 2-adic metric is defined. Showing that if $d(x,y)$ has a midpoint, then $x=y$

Let $\mathbb{Z}$ be the integers. Recall 2-adic metric $$ d(x,y) = \begin{cases} 0 & x=y \\ \frac{1}{2^{n}} & x \ne y\ \text{and}\ 2^{n} \text{is the largest power of 2 that ...
3
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0answers
29 views

Topological features (and / or definition) of homology

I am coming to grips with basic homological algebra as of late, in order to better understand my own subject, that of the study of language. The thing is, I have recently read in some handbook that ...
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2answers
30 views

Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
3
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2answers
38 views

Fundamental group of $\mathbb{R}^n\backslash \{0\}$

I am wondering about what the fundamental group of $\mathbb{R}^n \backslash \{0\}$ or more generally $\mathbb{R}^n \backslash U$ where $U$ is a subset of $\mathbb{R}^n$ for $n>1$. For $n=1$ I ...
2
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0answers
31 views

Question about contractible set .

Please if i have a contractible and closed set $A$ in $X$ thene $A$ is closed and there existe a continuous function $H:[0,1]\times A\rightarrow X$ such that $H(0,u)=u, H(1,u)=p\in X.$ If i ...
3
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0answers
32 views

Defining a topological relationship between two objects

I am looking for a mathematical definition/description of the following relationship between two objects. It's similar to a knot (as in topology) but between two objects. I've found a similar problem ...
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1answer
23 views

If X is limit point compact space,which is T1,then X is countably compact.

Countably compact means : every countable open covering contains a finite subcollection that covers it. Limit point compact means: every infinite set contained in it has a limit point. In T1 space ...
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+50

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
1
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1answer
22 views

Set of all real numbers with the Scott topology

It is known that a space $X$ is compact iff every net in $X$ has a cluster point. Let $\sum\mathbb{R}$ be set of all real numbers with the Scott topology. I know that $\sum\mathbb{R}$ is not compact. ...