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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13 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 ...
0
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1answer
24 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?
-1
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0answers
17 views

Help with general topology questions [on hold]

I am facing problem with this question proof can anyone help me on this. Thank You.
-3
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3answers
48 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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1answer
25 views

doubt with proof in genral topology

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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0answers
23 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 ...
0
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1answer
40 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

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
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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
22 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, ...
1
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0answers
41 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 ...
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1answer
13 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, ...
1
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1answer
24 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
16 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 ...
1
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1answer
27 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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0answers
9 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 ...
1
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2answers
38 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 ...
0
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0answers
23 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 ...
0
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0answers
16 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
20 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
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1answer
16 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
36 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
49 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 ...
1
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1answer
28 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
43 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
24 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. ...
3
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0answers
31 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
19 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 ...
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0answers
26 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
27 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 ...
2
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2answers
33 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 ...
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0answers
27 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 ...
2
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0answers
31 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
22 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, ...
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1answer
21 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. ...
0
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1answer
33 views

How to show that this set is closed in $\mathbb{R}^n$?

For an open set $\Omega\subseteq\mathbb{R}^n$, let $K_j$ be the set of points $x$ of $\Omega$ such that $\text{dist}(x,\partial\Omega)\geq1/j$ and $|x|\leq j$. Question : Why is $K_j$ closed ? ...
2
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1answer
31 views

$\mathfrak{Top}$ and injective objects

My question is very simple. Let $\mathfrak{Top}$ be the category of all topological spaces and continuous functions between them. Does such category have enough injectives? Is there a simple way to ...
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2answers
25 views

Induced subgroup of $\pi_1(S^1)$ by $p_n$

Consider the following covering map $p_n: S^1 \to S^1, z \mapsto z^n$. Why is the subgroup of $\pi_1(S^1)$ induced by $p_n$ isomorphic to $n\mathbb{Z}$? I know that $\pi_1(S^1) \cong \mathbb{Z}$ but ...
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1answer
30 views

There is no metric d,so that (Q,d) is a connected space [duplicate]

Can anyone prove this? There is no metric d,so that (Q,d) is a connected space Q are rationals.
3
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1answer
57 views

Locally Compact Spaces: Nonexamples

For Hausdorff spaces the following are equivalent: Every point admits a compact local base. Every point admits a compact neighborhood. Every point admits a precompact neighborhood. Every point ...
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3answers
42 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
2
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2answers
82 views

Construct a set of real numbers whose limit points comprise the set of integers $\mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only ...
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0answers
38 views

Cantor set--nowhere dense, complete

I can't figure out this out. Cantor set is closed in $\mathbb{R}$. $\mathbb{R}$ is a complete metric space. Every closed subset of a complete space is also complete; thus, so is the Cantor set. ...
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0answers
27 views

Separable spaces and functions that separate points

In a metric space, does existence of a function that separates points imply that the space is separable and conversely? I'm just a baby Rudin student. Thanks in advance for every hint.
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1answer
37 views

Can interior set or exterior set be empty?

I am trying to prove or disprove that if x is a boundary of S in R, then every ball B(x) contains both interior point of S and exterior point of S. I am trying to think of counter example, and one ...
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0answers
39 views

If $E$ is a closed set there exist a set $S$ such as $E=S'$

In "Elementary Real Analysis" by Thomson-Bruckner p.190 I did the following exercise: (we're working on $\mathbb{R}$ and elementary topology on that set) One of Cantor's early results in set ...
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1answer
41 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
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1answer
35 views

How does one see connectedness of a covering space?

Something can be proven about the loops (or their possible lifts?) in the base space which will ensure connectivity of the cover?
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2answers
45 views

Given $S \subset \Bbb{R}$, show $\textbf{int}(S)+\textbf{ext}(S)+\partial S =\Bbb{R}$

The way I proved it is that we knwo R is open so intR=R. For any point in IntS is inside of IntR and any point in ExtS is inside of IntR. any point that is neither intS nor extS is still inside of ...