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Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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What are the sets on which norm-closedness implies weakly closedness?

Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
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1answer
75 views

Closure of $\left\{\frac{\tan{n}}{n} | \, n \in \mathbb{N}\right\}$

I tried searching mathematical literature but I was unable to come across anything apart from the non-convergence of the sequence. Simply put: Define $\displaystyle A = \left\{ \frac{\tan{n}}{n} | \,...
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1answer
36 views

Connected set in $\Bbb R^2$

Let $$S:= \left\{\left(x, \sin\frac{1}{x}\right) \bigm\vert 0<x\le 1\right\}$$ denote a subset of the plane. Since $S$ is the image of the connected set $(0,1]$ under a continuous map, $S$ is ...
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1answer
13 views

equivalent of metric spaces and their topology [duplicate]

how we can give an example of two metrics on a space that induce the same topology , but are not equivalent!(so do we can generalize our definition about equivalent of metrics?)
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0answers
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sigma-compactness and $\omega$-cover

Let $X$ be a $\sigma$-compact space. An open cover $\mathcal{U}$ of $X$ is said to be an $\omega$-cover if for any finite subset $F$ of $X$ there is a $U\in \mathcal{U}$ such that $F\subset U.$ Now my ...
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2answers
55 views

Sierpiński space

Given a doubleton $X = \{0, 1\}$, the Sierpiński space is the ordered pair $(X, S)$ where $S = \{\emptyset, \{1\}, X\}$ is a topology on $X$. The Sierpiński space is the smallest example ...
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0answers
37 views

A set which is the closure of its interior points

I am trying to give a sufficient condition for a set in $\mathbb{R}^n$ which is the closure of its interior points. A prior, such a set has to be a closed set. A closed set in general is never the ...
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1answer
27 views

how many guards are needed to protect a king from an assassin in n-torus ($\mathbb R^n/\mathbb Z^n$)

Question: how many guards are needed to protect a king from an assassin if all of them are located on the $n$-torus $\mathbb R^n/\mathbb Z^n$? The location of the king and the assassin are known, ...
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0answers
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Equivalence of equicontinuity definitions in uniform space

Let $X$ be a compact Hausdorff topological space and let $G$ be a topological group acting continuously on $X$ from right. We know that $X$ admits a unique uniform structure $\mathscr F$. We call a ...
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2answers
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Conformal circles (A property of inversions)

I came across the following proposition in this book: If C is a circle in the Euclidean plane, iC is conformal, that is it preserves angles. Also, iC takes circles not containing the center of C to ...
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1answer
47 views

Every endomorphism $\psi$ of $\pi_1(S^1\times S^1, (1,1))$ can be expressed as $\psi = f_\ast$?

I have to prove the following: Using the canonical isomorphism $$\pi_1(S^1\times S^1,(1,1)) \cong \pi(S^1,1)\times \pi_1(S^1,1),$$ show that every endomorphism of the group $\pi_1(S^1\times S^1,(1,...
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1answer
21 views

Is $ G(x) = \nabla F(x)^* $ continuous when is $F$ continuously differentiable in Frechet's sense?

Let $F: X \to Y$ be a continuously differentiable function between banach spaces $X$ and $Y$(we also can assume $X$ is reflexive space) Let $x_0 \in X$, this tells us the derivative $\nabla F(x_0) \in ...
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0answers
28 views

Does $x_i \to x$ and $y_i^* \overset{w^*}{\rightarrow} y^*$ imply $ \nabla F(x_i)^* y_{i}^* \overset{w^*}{\rightarrow} \nabla F(x_0)^* y^* $

Let $F: X \to Y$ be a continuously differentiable function between banach spaces $X$ and $Y$. (we also can assume $X$ is reflexive space) Let $x_0 \in X$, this tells us the derivative $\nabla F(x_0) \...
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1answer
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Does $x_n$ posses a weak$^* -$ convergent sequence by banach-aluoghlo Theorem?

Let $X$ be a Banach space with the dual $X^\ast$. Let $\{x_n\}_{n=1}^{\infty}$ be a norm-bounded sequence in $X^\ast$, then can we claim that $x_n$ posses a weak$^\ast$-convergent subsequence by ...
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2answers
47 views

box topology vs product topology

Given a countably infinite family of topological spaces $(X_1,\tau_1),..,(X_n,\tau_n),...$, and their product $X$, I read that the box topology has as its basis the family: $$ B' = \prod_{i=1}^{\...
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1answer
41 views

Existence of compact/complete metric on countable set

Let, $X$ be a countable set. Which of the following are true? There exists metric $d$ on $X$ so that $(X,d)$ is complete. There exists metric $d$ on $X$ so that $(X,d)$ is not complete. There exists ...
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3answers
26 views

Does compactness of a subset use open sets from subspace topology for its finite subcover?

I read that a subset $S$ of a topological space $(X,\tau)$ is compact if for every open cover of $S$, there exists a finite subcover, i.e.: $$ S \subseteq \bigcup_{i \in J}O_i $$ for a finite $J$, ...
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0answers
49 views

Show $\mathbb{R}^2 - \mathbb{Q}^2$ is connected. [duplicate]

Show $\mathbb{R}^2 - \mathbb{Q}^2$ is connected. Now, I feel like showing that this is path connected is pretty simple, but showing that it is connected? The definition the we learned in class of a ...
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1answer
32 views

Star-Convex Set Centers Form Convex Set

I have the following proof that all centers of a star-convex set $U\subset\mathbb{R}^n$, $Z(U)$ form a convex set: Suppose $Z(U)$ is not convex, then there are elements $a,b\in Z(U)$ such that the ...
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1answer
43 views

Prob. 6, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: The image of a normal space under a closed continuous map is also normal

Here is Prob. 6, Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $p \colon X \to Y$ be a closed continuous surjective map. Show that if $X$ is normal, then so is $Y$. [Hint: If $...
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Do all non-orientable have to “close” in some direction?

I am not a connoisseur in the topic of surfaces, the two non-orientable surfaces I know are the Möbius strip and the Klein bottle. Intuitively I understand that for a surface to be non-orientable, ...
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2answers
62 views

Suppose that $f: \mathbb R^n \to \mathbb R^n$ is a bijection and $n\geq2$. Can $f$ send every open set onto non-open set?

Suppose that $f: \mathbb R^n \to \mathbb R^n$ is a bijection and $n\geq2$. Can $f$ send every open set onto non-open set? I do not know what exactly to write about this problem. Did I try anything? ...
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0answers
24 views

Some details about the reverse direction of Kakutani's theorem

I'm working on understanding the proof of the reverse direction of Kakutani's Theorem: Banach space is reflexive if and only if its closed unit ball is weakly compact The proof is given in Royden ...
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2answers
26 views

The complement of a bounded region (the exterior region) is also a bounded region?

My textbook, An Introduction to the Mathematical Theory of the Navier-Stokes Equations by Galdi, says that $\Omega$ is a type of region where flow occurs. To this end, it distinguishes between the ...
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0answers
41 views

Checking if two metrics induce the same topology

We got two metrics $d$ and $d'$, specified for the set of natural numbers by using formulas $d(k,n)=\vert{k^2 - n^2}\vert$ and $d'(k,n)=\vert{\frac{1}{k} - \frac{1}{n}}\vert$. Do they induce the same ...
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0answers
14 views

Why is the space of 2D lattices topologically 4D, if Gram matrices are 3D?

There are several good articles on the space of 2D lattices, usually represented using Eisenstein series and the invariants $G_4$ and $G_6$. Here are some: http://www.stevejtrettel.com/the-space-of-...
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0answers
38 views

Inverse Limit of Finite Direct Product of Groups/TopSpaces

I’m wondering if the inverse limit of a finite direct product of groups $G_{i,j}$ or topological spaces $X_{i,j}$ is isomorphic to the direct product of the inverse limits of $G_{i,j}$ or $X_{i,j}$, ...
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2answers
50 views

What machinery is used in Hatcher Thm 2.26?

This is related to Hatcher Thm 2.26's proof. If non empty open subsets $U\subset R^n, V\subset R^m$ are homeomorphic, then $m=n$. Pf. $x\in U$, then $H_k(U,U-\{x\})=H_k(R^n,R^n-\{0\})$ which is $Z$...
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1answer
55 views

If $X$ is a countable set, does there exist a metric that makes it compact?

If $X$ is a countable set, does there exist a metric that makes it compact? I know there is always a metric that does not make it compact and that there are examples of compact countable spaces but I ...
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2answers
17 views

Show that if $ X $ is regular, then for every neighborhood $U$ of $ x \in X $ there exists a neighborhood $ V $ such that $ \overline{V} \subset{U} $

Show that if $ X $ is regular, then for every neighborhood $ U$ of $ x \in X $ there exists a neighborhood $ V $ such that $ \overline{V\mkern-0.5mu} \subset{U} $. Given $ x \in X $ and a ...
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1answer
37 views

Continuity on the one-point compactification of $R^n$

We have seen that the one-point compactification of $R^n$, given as $R^n\cup \{\infty\}$ for a point not in the set, and denoted $\hat{R^n}$, is homeomorphic to the sphere $S^n$. I equip $\hat{R^n}$ ...
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2answers
44 views

This bijective map is continuous?

Let $f:X\rightarrow X$ a bijective map between topological spaces (the same space X). A priori not known to be continuous. If we know $f\circ f=id$ does it mean that $f$ has to be continuous map and ...
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1answer
36 views

a question about star compact space

Definition 1: ‎If ‎‎$‎X‎$‎ is a topological space and ‎‎$‎‎\mathcal{U}‎‎ ‎‎$‎ is a family of subsets of ‎‎$‎X‎$‎, then the star of a subset ‎‎$‎A ‎\subseteq‎ ‎X‎$‎ with respect to ‎‎$‎\mathcal{U}‎‎‎$‎...
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1answer
42 views

Proposition regarding finite dimensional algebras

In the book 'Matrix Groups' by Andrew Baker the following proposition is given without proof. $A$ is finite dimensional k-algebra. Then 1) The function $\chi : A^{\times} \rightarrow \...
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1answer
23 views

Line with two origins: Homeomorphism

Let $A$ and $B$ be two points not on the real line $\mathbb{R}$. Consider the set $S=(\mathbb{R}\setminus\{0\})\cup\{A,B\}$. For any two positive real numbers $c,d$ define $$I_A(-c,d)=]-c,0[\cup\{A\}\...
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1answer
30 views

How to show that $\mathbb{N}$ is a Baire space [duplicate]

I read that a topological space $(X,d)$ is a Baire space if for every sequence $\{X_n\}$ of open dense subsets of $X$, the set $\bigcap_{n=1}^{\infty}X_n$ is also dense in $X$. Since every complete ...
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3answers
60 views

Countable product of metric spaces is metrizable [General Metric]

I know that if we have a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$ then $X=\Pi^{\infty}_{n=1}X_n$ is a metric space with metric $\rho((x_n)_{n \in \mathbb{N}},(y_n)_{n \...
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1answer
20 views

Showing these definitions of 'perfectly normal' spaces are equivalent

It seems to me that the first definition given is just strictly stronger than the second definition. I've tried to construct a function, using definition 2, that satisfies definition 1 but haven't ...
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1answer
29 views

Showing that the real projective space is Hausdorff using matrices and linear algebra

I am trying to follow the proof in Loring Tu's book (An introduction to smooth manifolds, 2nd edition, p.79) to show that the real projective space is Hausdorff. A snippet of the proof is shown below. ...
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0answers
25 views

is A a closed set? $A= \lbrace x_1,x_2 \in \Bbb R^2: 3x_1^4 +7x_2^3+x_1^2-x_2^2\ge30 \rbrace$ [on hold]

let be $A= \lbrace x_1,x_2 \in \Bbb R^2: 3x_1^4 +7x_2^3+x_1^2-x_2^2\ge30 \rbrace$ prove that A is closed any hints
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0answers
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“Prove that ∂A ∪ Int(A) = Cl(A)” [duplicate]

The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. I have very little knowledge of set theory and proofs, so I'm not sure how to prove this. As ...
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1answer
54 views

Does $x_n \overset{w}{\rightarrow} x$ and $Tx_n \to y$ imply $y = Tx$

Let $T: X\to Y $be a bounded linear operator between reflexive Banach spaces $X$ and $Y$ , and let $\{x_n\}$ be sequence in $X$ with property $x_n \overset{w}{\rightarrow} x$ and $Tx_n \to y$. Can we ...
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1answer
33 views

let $P \subseteq T$ ,then which of the following option are correct?

Given $T $ be a topological space and let $K_1 , K_2$ be two dense subsets of $T.$ let $P \subseteq T$ ,then which of the following option are correct ? $1.$ $ P \cap K_1 $ is dense in $P$ $...
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0answers
32 views

Proving the Non-Emptiness of an Intersection of Neighborhoods in Euclidean k-Space

This is obviously true but I can't figure out how to prove it: Let $r>0$. Let $\varepsilon>0$. Let $k\in\{1,2, \ldots\}$. Let $a\in\mathbb{R}^k$. Let $p\in\mathbb{R}^k$ satisfty $|p-a|=r$ ...
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2answers
37 views

Prove that $\partial A = \mathrm{Cl}(A) \cap \mathrm{Cl}(X − A)$

My textbook doesn't give me the fact that $\partial A = \mathrm{Cl}(A) \cap \mathrm{Cl}(X − A)$. We're asked to prove it. I'm given $\partial A = \mathrm{Cl}(A) − \mathrm{Int}(A)$. In a previous ...
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1answer
32 views

Example 3.5. - Chapter 0, Do Carmo's Riemannian Geometry

The following map from $\alpha : \mathbb{R} \to \mathbb{R}^2$ is not an embedding $$ \alpha(t) = \left\{ \begin{array}{ll} (0,-(t+2)) & t \in (-3,-1) \\ \text{regular curve} & t \in (-1,-1/\...
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2answers
27 views

If $B$ is a basis for a topology $T$, does $B$ necessarily generate $T$?

Let $(X,\tau)$ be some topological space. Munkres defines a basis $\mathcal{B}$ of $\tau$ as a collection of subsets of $X$ such that: $\mathcal{B}$ covers $X$, and given $B_1, B_2 \in \mathcal{B}$, ...
0
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1answer
26 views

Definition of disconnected set in topology

I'm reading through wikipedia for a rigorous definition of disconnected topological space, which is the same as the one given by Munkres. A topological space $X$ is said to be disconnected if it is ...
2
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1answer
47 views

Prob. 3, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Every order topology is regular

Here is Prob. 3, Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Show that every order topology is regular. First of all, here are some relevant definitions. Ordered Set: Let $...
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2answers
50 views

Counter example for Baire's Theorem

Theorem: Let $(X,d)$ be a complete metric space, and let $D_n, n\in \mathbb N$ be open, dense subsets of $X$. Then also $\bigcap_{n\in\mathbb N} D_n$ is dense in $X$. This statement is false if $X$ ...