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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10 views

LCA - groups under continuous homomorphisms

can someone help me out with this question? LCA stands for Locally compact Hausdorff abelian group. The question is posted in the attached image
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2answers
35 views

Are continuous functions with compact support bounded?

While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded continuous ...
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2answers
22 views

Equivalence to being a topological group

Just some notation I am using: A topological group $G$ is a group with a topology such that $o : G^2 \to G : (x,y) \mapsto xy$ and $inv : G \to G : x \mapsto x^{-1}$ are continuous in the ...
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2answers
39 views

Prove $f$ is Lipschitz on $K$

Let $f:\mathbb{R}^d\to \mathbb{R}$ such that it's partial derivatives are continuous. Let $K\subseteq \mathbb{R}^d$, a bounded set. Prove that $f$ is Lipschitz on $K$. My work: Since $f$'s ...
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23 views

What is the definition of the following concepts and how I can characterize each concept.

What is the definition of the following concepts and how I can characterize each concept. A set $A\subseteq 2^{\omega}$ is Lebesgue measurable zero if ? A set $A\subseteq \omega^{\omega}$ is ...
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0answers
23 views

Fibre of a local homeomorphism can be covered by disjoint open sets.

Let $f\colon X\rightarrow Y$ be an open local homeomorphism and $y\in Y$. Do there exist pairwise disjoint open neighborhoods $U_x$ for $x\in f^{-1}(y)$? If not, what would be mild topological ...
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1answer
34 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
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1answer
17 views

Hausdorff Lindolef Space is Regluar?

I think we can use same argument for saying regluar Lindolef space is normal to prove Hausdorff Lindolef space is regluar. But, I didn't heard about this proposition. What is the problem of using same ...
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26 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
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2answers
25 views

Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...
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3answers
54 views

$x^2+y^2<1, x+y<3$ is open or closed?

I'm trying to figure out if $$\{x^2+y^2<1, x+y<3|(x,y)\in \mathbb R^2\}$$ is open or closed. I tried to imagine this set. It looks, for me, as a 'pizza', or a circular sector, which have two ...
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1answer
45 views

Open sets in the product topology

Let $\{X_{\alpha}\}_{\alpha\in A}$ be a family of topological spaces. The product topology on $X=\prod_{\alpha\in A}X_{\alpha}$ is the weak topology generated by the coordinate maps ...
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40 views

Fundamental group two torus minus a single point?

So, if I take one torus and take of one single point, what will be its fundamental group? I think that one single point will not change the topology in this sense. Or will? If yes, how?
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1answer
24 views

Show that a regular space, under a “new” topology, is Tychonoff

I have this problem that I can't solve it: Problem: Let $(X, \tau)$ be a regular space. Let $\tau_\delta = \{A \subseteq X \, \: \, \forall a \in A \, \, \exists \, W \subseteq X \textrm{ such that } ...
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49 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
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1answer
60 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
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1answer
38 views

Is this argument invalid?

So, what I was trying to prove is: Let $\pi : X \to Y$ the quotient map such that $\pi^{-1}(\{y\})$ connected for all $y \in Y$. Then $X$ is connected. Suppose yet that $X$ is locally connected. ...
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1answer
50 views

Why is $f(x,y)$ said to be discontinuous at $(0,0)$?

Why is $f(x,y)=\begin{cases} \frac{x^2y}{x^4+y^2}, & \text{if $(x,y)\neq (0,0)$}\\[2ex] 0, & \text{if $(x,y)=(0,0)$} \end{cases}$ said to be discontinuous at $(0,0)$? I am supposed to show ...
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1answer
27 views

Open (closed) sets of a locally compact space.

Let $X$ a locally compact space. How do I show that if $A$ is a open (closed) set in $X$ then $A$ is locally compact? Thank you very much.
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3answers
34 views

If $K\in \Bbb{R}^n$ is compact, $\sup_{x,y\in K}|x-y|=\max_{x,y\in K}|x-y|$.

Suppose $K\in \Bbb{R}^n$ is compact. Let us denote $D=\sup_{x,y\in K}|x-y|$ as $K's$ diameter. Prove there exist $a,b\in K$ such that $D=|a-b|$ i.e, that the suprimum is the maximum. I know there ...
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1answer
38 views

Mathematical pre-requisites to read History of Topology by I. M. James [on hold]

What are the necessary mathematical pre-requisites to read History of Topology by I. M. James?
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28 views

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. [duplicate]

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. I have a problem proving the direction according to which $A$ is compact. First direction I said: If $A$ is ...
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1answer
40 views

Lexicographic Orderings of Aronszajn Trees are Aronszajn Lines

I am trying to prove that every lexicographic ordering of a Aronszajn tree is a Aronszajn Line. If $T$ is a tree, a lexicographic ordering of $T$ is defined as follows: For each ...
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2answers
40 views

(A question regarding:) the graph associated with an open cover of a topological space.

Let $X$ denote a topological space and suppose that $\mathcal{O}$ is an open cover of $X$. Assume $\emptyset \notin \mathcal{O}.$ (Thanks Niels!) Now make $\mathcal{O}$ into an (undirected) graph as ...
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1answer
63 views

Density of sin(k) where k is an integer [duplicate]

Consider the set $A=\{\sin k:k\in\mathbb Z \}$. I want to know whether this set is dense in $[-1,1]$. I have a hunch that this problem can somehow be reduced to the approximation of $\pi$ using ...
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0answers
33 views

Connected set imply continuous boundary [on hold]

Is it true that a connected and bounded set of $\mathbb{R}^2$ has a boundary that can be parametrized by a continuous mapping?
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0answers
32 views

Product Topology and Borel-$\sigma$-algebra

Let $S=\left\{1,2,...,n\right\}$ be equipped with discrete topology and let $X=S^{\mathbb{Z}}$. Then the so-called cylinder sets $$ [s_0,s_1,...,s_m]_n:=\left\{x\in X: \forall 0\leqslant i\leqslant ...
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0answers
39 views

Proof of Supporting Hyperplane Theorem from basic definitions.

My purposes in posting this question are twofold. First, I would like to have a lemma which I have proven on the way to proving the Supporting Hyperplane Theorem checked for rigor (zero tolerance for ...
4
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1answer
66 views

Aronszajn lines

Exercise 32 of chapter 2 of Kunen (1980) tells me to show that there exists a total ordering with no $\omega_1$ strictly increasing/decreasing sequencies such that every separable subspace is nowhere ...
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3answers
71 views

Which of $(-\infty,\infty]$ and $[-\infty,\infty]$ is homeomorphic to $S^1$?

Is it correct to say that $(-\infty,\infty]$ is homeomorphic to $S^1$? or it is $[-\infty,\infty]$? (considering standard topology). Would you please provide some explanation or better a rigorous ...
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2answers
66 views

What is the difference(s) between $(a,\infty)$ and $(a,\infty]$?

I am studying H. L. Royden's Real Analysis which includes some introduction to Measure Theory; and I encountered $(a,\infty]$ instead of $(a,\infty)$ for the first time! What is the difference(s) ...
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3answers
72 views

In an Euclidean $\mathbb R^n$ space, is every compact set an open set? Is it possible to have sets that are both open and bounded?

I know that compact sets are the ones that are both bounded and closed (Heine-Borel Theorem), but since closed and open are not opposites, I cant see if and how a compact set, or a bounded set, can be ...
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0answers
32 views

Topological spaces with the same underlying set and basis. [on hold]

Let $(X; T_1)$ and $(X; T_2)$ be topological spaces with the same underlying set. Let $B_1$ and $B_2$ be bases for $T_1$ and $T_2$ respectively. Then $T_1$ = $T_2$ if and only if $B_2\subseteq ...
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3answers
142 views

Show the following set is connected

For any $x \in \Bbb R^n$ how do I show that the set $B_x := \{{kx\mid k \in \Bbb R}$} is connected. It should also be concluded that $\Bbb R^n$ is connected. I was thinking of starting by assuming ...
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0answers
27 views

Is $\tau$ a topology on $\mathbb{R}^2$? where the elements of $\tau$ are $\emptyset$ and the complements of finite sets of lines and points [on hold]

Prove that ($\mathbb{R}^2$,$\tau$) is a topological space where the elements of $\tau$ are $\emptyset$ and the complements of finite sets of lines and points I don't know how to prove the second ...
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0answers
17 views

Find the boundary and interior of subsets of $\mathbb{R}^2$ [on hold]

Find the boundary and interior for each of the following subsets of $\mathbb{R}^2$: -$A =\{ (x,y) \in \mathbb{R}^2 \colon y = 0 \}$, -$B = \{ (x,y) \in \mathbb{R}^2 \colon x > 0, y \not=0 \}$.
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1answer
41 views

Properties of closure and example

Let $A_1, A_2, A_3, \dots$ be subsets of a metric space. If $B=\bigcup_{i=1}^\infty A_i$, prove that $\overline{B}\supset \bigcup_{i=1}^\infty \overline{A_i}.$ Show, by an example, that this ...
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0answers
14 views

Construct an example in which $x$ is $\tau_1$-accumulation point of a subset $A$ of $X$ but It is not $\tau_2$-accumulation point of $A$

Let $\tau_1$ and $\tau_2$ be a topologies on a set $X$ with $\tau_1 \subset \tau_2$ Construct an example in which $x$ is $\tau_1$-accumulation point of a subset $A$ of $X$ but It is not ...
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3answers
51 views

Proof $\{(x,y,z)|4x^2+9y^2+16z^2<1\}$ is an open set

In order to prove that the points $(x,y,z)$ such that $$4x^2+9y^2+16z^2<1$$ form an open set, I tried this: Pick a generic point of the ellipsoid, lets say $$4x^2+9y^2+16z^2$$ Now, I'll form ...
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3answers
183 views

Is a ball noncompact?

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
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1answer
33 views

Topology (Basis)

A basis for a topology is defined as the subcollection of the topology such that every member of the topology can be expressed as the union of members of that subcollection. But if the basis doesn't ...
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0answers
19 views

Is the set of limitpoints of a set always closed? [duplicate]

If $E$ is a subset of a topological space and $E'$ denotes its set of limitpoints then I can prove that $E'$ is closed under the extra condition that the topological space is $T_1$. For a proof see ...
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1answer
32 views

About expansive homeomorphim

We say $(X,f)$ is expansive if there is $c(f)>0$ such that if $d(f^{n}(x), f^{n}(y))< c(f)$ for every $n\in Z$ then $y=x$. Let $(X,f)$ is expansive with constant $c(f)$ and for infinite set ...
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1answer
16 views

What 'limit point' means in $\Bbb Z_+\times {\{a,b}\}$?

Here was a question, but raised another question of meaning of a neighborhood. According to the answer: [Let] $Y=\{a,b\}$. If $S$ is a subset of $\Bbb Z_+\times Y$ and $(n,a)\in S$, then $(n,b)$ ...
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0answers
33 views

The set of all limit point of a set

Let $E'$ be the set of all limit points of a set $E$. Do $E$ and $E'$ always have the same limit points? Proof: $E''\subset E'$ because $E'$ is closed. But inclusion $E'\subset E''$ is false. We ...
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1answer
70 views

Proof that $f$ is an isometry

Let $D$ be the set of points in $\Bbb R^2$ such $\lvert p \rvert\leq1$, and let $f: D\rightarrow D$ be a surjective function, satisfying this relation: $\lvert f(p)-f(q)\rvert \leq \lvert p-q\rvert$, ...
3
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89 views

A Question Regards Quotient Topology and Vector Space

Let $V$ be a $n$ dimensional vector space,Consider the topological space $(V, \mathcal{T}_v)$, where $\mathcal{T}_v$ is standard topology on $V$. Standard topology on $V$ is defined by ...
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32 views

homeomorphism as a result of other homeomorphisms

If $$B = \bigcup_{R>0} B_R$$ and all the identities $$\operatorname{id}_R : (B_R,d_1) \rightarrow (B_R,d_2)$$ for $R>0$ are homeomorphisms, then is $$ \operatorname{id} : (B,d_1) \rightarrow ...
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2answers
65 views

Can we define $ℝ^A$ where A is uncountable?

The question is pretty straightforward. How can we define the expression $ℝ^A$ when $A$ is an uncountable set? For example what is defined by forms such as $ℝ^ℝ$ or $ℝ^ℂ$? If $A$ is countable,then ...
4
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2answers
65 views

Is a continuous function $f : \mathbb{Q}\to\mathbb{Q}$ always bounded on a closed interval?

Can a function $f : \mathbb{Q} \to \mathbb{Q}$ that is continuous on an interval $[a,b]$ not be bounded on $[a,b]$? I'm asking this because in Spivak's Calculus, the "Boundedness Theorem", which ...