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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2
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31 views

Basic Topology: Limit Points

Please see the below and let me know if my logic is not correct for this problem. Thank you. Let $A = (3,4) \cup \{5,6\}$. In the standard topology, prove the following: (a) There is a point ...
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2answers
23 views

The closure of a subset in finer topology is always subset of the closure of that subset in the coarser one.

On the Appendix A of Naber book "The Geometry of Minkowski Spacetime" there is a claim in Lemma A.3.3. It says that if we have a set (says $M$) endowed with two different topology says $(M,O_A)$ and $(...
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1answer
36 views

Triangle inequality in $\kappa $ metric space where $\kappa = 2^n $

$X$ is a $\kappa $ - metric space if $$d: X\times X \rightarrow \mathbb R$$ satisfies the following : $$a)\ \ d(x,y)\ge 0;\\b)\ \ d(x,y)=d(y,x);\\c)\ \ d(x,y)=0\ \ \iff\ x=y;\\d)\ \ d(x,z)\le \...
1
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1answer
49 views

Is my proof of the fact that the product of two compact metric spaces is compact correct?

Sometimes ago I have posted this question. After sometime of working I think that I have found out a different proof (not "purely topological"). I didn't post it there as an answer because (1) the ...
0
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1answer
22 views

Limit Points and Closures in Topology

I am in an introductory Topology course and the text is not very helpful. One of the example questions state the following: Let A = [0, 1) u (1,2) be a subset of (R,U). Then find the following: Int(...
0
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2answers
31 views

Difference in definition of a cover

Let $X$ be a topological space, Ive seen some define a cover, as a collection, $A$, of sets such that $X=\bigcup A$, while others use the condition $X\subseteq \bigcup A$. Why the difference? My ...
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3answers
40 views

Justify why all constant functions $f: X \rightarrow Y$ are continuous.

I have read that if we let $(X, \tau_x)$ and $(Y, \tau_y)$ be topological spaces, and $f: X \rightarrow Y$ is constant, then f is continuous. Can anyone please explain why? I assume we must look at ...
1
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1answer
58 views

Topological proof of the compactness of product metric space

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric spaces (see the definition here). Then show that the product metric space $(X\times Y,d_{X\times Y})$ is also compact. Now this can be done ...
3
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0answers
40 views

Extension of Kakutani's fixed point theorem.

Can the Kakutani's fixed point theorem's be extended to say that there exists a fixed point inside the set (not on boundary)(I am not sure how to formally state this). For a $n$-dimensional compact, ...
1
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1answer
42 views

Prove that $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$ is not compact

Let $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$ Show that $K$ is not compact
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1answer
49 views

Is $K^{n}$ Zariski Hausdorff when $K$ is a finite field?

Assume that $K$ is a finite field. Is it true to say that $K^{n}$ is a Hausdorff topological space with Zariski topology?
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0answers
64 views

Isomorphism between pairs of dual vector spaces

I am trying to self-learn functional analysis via the book Topological Vector Spaces by Robertson and Robertson and I am stuck at the following which they state without proof. This is stated on page ...
1
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3answers
71 views

Given a function $f: X \to Y$, if $X$ is compact, prove the graph $g = (x, f(x))$ is compact in $X\times Y$

Given a function $f: X \to Y$, and graph of $f, g = \{(x, f(x)): x\in X\}$ in metric space $X\times Y$ (a) Suppose that $X$ is compact. Prove that $f$ is continuous if and only if $g$ is a compact ...
1
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1answer
21 views

$X$ is completely regular iff it carries the initial topology w.r.t. $C(X,\mathbb{R})$

I've read that A topological space $X$ is completely regular iff it carries the initial (weak) topology w.r.t. $C(X,\mathbb{R})$ where $C(X, \mathbb{R})$ is the set of all bounded real-valued ...
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2answers
42 views

Prove that $S \times S$ is not Lindelöf

Prove that $S \times S$ is not Lindelöf, where $S$ denotes the Sorgenfrey line. I know that $S$ is Lindelöf even though it is not second countable. I would believe that the product would also be ...
1
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1answer
40 views

Antipodal gluing of a disk is homeomorphic to a disk.

Given $D=\{(x,y)\in \mathbb{R}^2 \mid x^2+y^2\leq 1\}$, if I define $\sim $ as $(x,y)\sim (x',y') \iff x'=-x, y'=-y$ is the gluing of points across the line $y=-x$. The same way that the antipodal ...
4
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1answer
53 views

If $M$ is a connected manifold, does $M\setminus\{p\}$ have finitely many components?

Let $M$ be a connected manifold and $p\in M$. Is it true that $M\setminus\{p\}$ has only finitely many connected components? (We can also suppose $M$ is compact if that helps.) I think this is ...
1
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1answer
57 views

Geometric xample of a one object cover which does not satisfy the sheaf condition?

For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self intersection)...
0
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1answer
31 views

Prove that the interior of a line segment joining two points on a sphere lies strictly inside of the sphere

Practically if $|x|=|y| = r$ then $|(1-t)x + ty| < r$ for $0 < t < 1$. But $|(1-t)x + ty| \leq (1-t)|x| + t|y| = r \Rightarrow |(1-t)x + ty| \leq r$ whats is the argumentation for this ...
0
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1answer
55 views

Prove that there is no continuous surjection from $S^n$ to $\mathbb{R}_n$

Let $S^n = \{(x_1, . . . , x_{n+1}) \in \mathbb{R}^{n+1} \mid \sum_{k=1}^{n+1} x_k^2 = 1\}$. Prove that there is no continuous surjection $f : S^n \to \mathbb{R}^n$.
0
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1answer
39 views

Basic Topology on $\mathbb{R}^{n}$

Given $a \neq b$ at $\mathbb{R}^{n}$ determine $c$ in $[ab]$ such that $c \bot (b-a)$. Conclude that for all $ x \in [ab]$, $x \neq c$ $|c| < |x|$. Okay, i could find the form o $c$, if $<c,b-a&...
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3answers
48 views

Justification that (a,b] in the standard topology is not closed?

I was wondering how to justify that $(a,b]$ is not closed in the standard topology on R. I know that the definition of a closed set is one whose complement is open. So then we have that R - (a,b] = $...
0
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1answer
25 views

Path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$

I am trying to find a path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$ where $t\in[0,1]$ $\alpha$ and $\beta$ are path homo topic if they have the same endpoints, $p, q$ and $\exists ...
0
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1answer
26 views

Proof for separated sets

Let A and B be separated subsets of $R^k$, suppose $a∈A, b∈B$,and define $p(t)=(1-t)a+tb$ for $t ∈R^1$. Put $A_0=p^{(-1)} (A), B_0=p^{(-1)} (B)$. [Thus $t ∈A_0$ if and only if $p(t)∈A$.] Prove ...
1
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1answer
19 views

Close graph of multivalued function

Assume that $X,Y$ are two closed sets in $R^n$ (with the induced euclidian topology) and let $f:X \rightarrow Y$ be a multivalued function. Assume also that the graph of $f$ is a closed subset in $X \...
4
votes
4answers
41 views

Is it true that $F\cap \overset{\circ}{E}=\emptyset\implies F\cap \overset{\_}{E}=\emptyset$?

Let $E,F\subset \mathbb{R}^n$ where $F$ is open and $E$ is arbitrary. Is it true that: $$ F\cap \overset{\circ}{E}=\emptyset\implies F\cap \overset{\_}{E}=\emptyset $$ Intuitively I think this is true,...
0
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1answer
22 views

limit points of ${x}$ in a $T_0$ space

Let $X$ be a $T_0$ topology space. How do I prove that for every element $x\in X$ the set of limit points of ${x}$ is a union of closed sets?
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0answers
14 views

discrete topology on simple metric space.

Given a set X, define d(x,y)=1 if x is not equal to y and d(x,y)=0 if x=y Then, the topology induced by metric d is discrete topology. Discrete topology is the collection of all subsets of X. If ...
0
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2answers
64 views

How to show $f(x)=x^{2}$ is a closed map?

The actual question was $\\$ "Show $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=x^{2}$ is closed but not open." For the latter one, $f((-1,1))=[0,1)$ so $f$ is not an open map, but I have ...
0
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1answer
35 views

Are representables on the étale site on topological space sheaves?

The gluing lemma says representables on the classical site on $\mathsf{Top}$ are sheaves. Basic scheme theory says the same is true for the small Zariski site of a scheme. Are representables on the ...
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0answers
18 views

How to express a measurable function equal to it's conditional expectation?

I'd like to know if I'm expressing this correctly. Apologies in advance: topology and I have never been well acquainted. Let $(m,\mathcal B, X)$ be a measure space, and $\mathcal H \subset \mathcal ...
1
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1answer
28 views

If $ \left\{ A_n : n \in \mathbb{N} \right\} $ is a *collection* of compact sets so that every finite subcollection has a nonempty intersection,

Is $\bigcap^\infty_{n=1} A_n$ nonempty? Does the same apply for closed sets? This question came up in review today in my Analysis I class, but no one was able to answer it before the period ended. ...
6
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3answers
106 views

Continuous bijection from $(a,b) \to S^1$?

This question started bothering me after working on an exercise. I know that there cannot be a contiuous bijection $S^1 \to (a,b)$ because if there was it would be a homeomorphism but $S^1$ and $(a,b)$...
0
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2answers
42 views

The set X is the complex numbers in cofinite topology

I have to check if $X\backslash\{0\}$ is homeomorphic to $X \backslash \{n|n\in\mathbb{N}\}$. Here I am considering $X\backslash\{0\}$ and $X \backslash \{n|n\in\mathbb{N}\}$ in subspace topology.
0
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1answer
101 views

How many fixed points must $f$ have in the disk? [closed]

Let $\Omega$ be an open subset of $\mathbb{C}$. Assume $f \in H(\Omega)$, $\Omega$ contains the closed unit disk, and $|f(z)| < 1$ if $|z| = 1$. How many fixed points must $f$ have in the disk?
0
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2answers
45 views

Why if $p (x_1, \dots, x_n)$ is polynomial on $\mathbb{R}^n$, then $p (x) \neq 0$ is satisfied by open dense set?

I have problems in seeing what exactly is the all point of first category and second category sets. Finally, I've found a reference (Bredon's "Topology and Geometry") that introduces the topic in a ...
1
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1answer
36 views

clarification about one point compactification

Munkres in his book Topology writes $X$ is locally compact Hausdorff iff there exists a space $Y$ such that (1) $X$ is a subspace of $Y$,(2) the set $Y-X$ consists of a single point if (3)$Y$ is a ...
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1answer
18 views

Show$d : x, y \in E \to \|x − y\|_E$ is a distance on $E$ and every open ball is the image of the unit ball under an affine map then check closure.

A norm $\| · \|$ $E$ on a $\Bbb R$-vector space E is a function from E to [0, ∞) such that: $\forall x \in E$ we have $\|x\|_E =0 \iff x=0$, $\forall x\in E$and$ \ λ \in \Bbb R$ we have $\|λx\|_E |...
9
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1answer
49 views

What is the quotient space of $\mathbb{Z}$-indexed $\{0,1\}$-sequences with resprect to shifts?

Let $V=\{0,1\}^{\mathbb{Z}}$ be equipped with product topology. For $k\in\mathbb{Z}$ let $T^k:V\rightarrow V$ be the shift-by-$k$-operator, so $$T^k((x_j)_{j\in\mathbb{Z}}):=(x_{j+k})_{j\in\mathbb{Z}}....
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0answers
15 views

Formal definition of “perimeter” of 3-dimensional polyhedron (also projection v. transformation v. mapping confusion)

One of my students was trying to use the "perimeter" of one of the square faces from a square cuboid to calculate other measurements. From my understanding, perimeter is only defined for polygons. ...
0
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2answers
54 views

Is every compact space hereditarily Lindelöf?

All spaces are assumed Hausdorff. We call a topological space compact if every open cover has a finite subcover. We call it Lindelöf if every open cover has a countable subcover, and hereditarily ...
0
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1answer
59 views

Compute the winding number of $\alpha(z)=4z^4+2z^2+1$ and $\alpha(z)=6z^2+7z+2$

The winding number of $\gamma$ about $0$ is given as: $w(\gamma, 0)=\frac{arg(\gamma(1))-arg(\gamma(0))}{2\pi}$ $\gamma : [0,1] \rightarrow \mathbb{C}-\{0\}$ is a loop in $\mathbb{C}$ not passing ...
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0answers
27 views

Question about limit points with continuous maps. What have I done wrong?

I'm trying to do this exercise: Suppose that $f:X \rightarrow Y$ is continuous. If $x$ is a limit point of the subset $A$ of $X$, is it necessarily true that $f(x)$ is a limit point of $f(A)$? I'...
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0answers
57 views

Computing fundamental group of the complement to three infinite straight lines, and of complement to $S^1 \cup {Z} $

Question 1: Find the Fundamental group of the complement to three infinite straight lines that have no common points in $\mathbb{R^3}$ Question 2 Compute the fundamental group of the complement of $...
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0answers
31 views

Morphisms of action group and continuity

Let $\mathbf{G}$ be a profinite group, $\mathbf{X}$ and $\mathbf{Y}$ two finite topological spaces with a continuous action group of $\mathbf{G}$ on them. If $\mathbf{G}$ is finite, is this true that ...
3
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1answer
65 views

Compact but not measurable [closed]

Does there exist a compact subset of $\Bbb{R}^n$ (with the usual topology) which is not Lebesgue measurable?
0
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1answer
24 views

Show any continuous function $f: \mathbb{\bar N} \to (\mathbb{R}, \tau_c)$ is eventually constant

Show any continuous function $f:\mathbb{\bar N} \to (\mathbb{R}, \tau_c)$ is eventually constant. Preliminaries: $\tau_c = \{U \subset \mathbb{R} \mid U = \emptyset \vee U^c : \text{countable} \}$...
0
votes
1answer
35 views

Open Maps between topological spaces

Let $X,Y$ be topological spaces. A map $f$ from $X$ to $Y$ is open if for every open set $U$ in $X$ , $f(U)$ is open in $Y$. Let $X= \mathbb{R}$ and $Y= \mathbb{S^1}$ and $f: t \rightarrow e^{2\pi i ...
0
votes
1answer
21 views

Determinate the quotient topology

I was trying to find the quotient topolgy for the next example: Let R be the real numbers with the usual topology ($\tau$) and define the relationship $\mathcal{R}$ over R as follows, a $\mathcal{R}$...
2
votes
1answer
40 views

Decimal expansions and topological connectedness

I'm a bit confused by the following footnote from Moschovakis's Notes on Set Theory, p. 135fn24 (in the note, $\mathcal{N}$ denotes the Baire space). The puzzling part is in bold: One may think of ...