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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1answer
14 views

Kelly's topology: Compact spaces and finite intersection property

Lemma A topological space is compact iff each family of closed sets which have the finite intersection propertyhas a non-void intersection. I've proved the same result in another way but i really ...
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24 views

When is a connected space, path connected?

Let $X$ be a connected topological space. When is $X$ path connected? Is the Hausdorff property enough? Is it too much?
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0answers
13 views

Prove that a complete field defines a partition of a set

Let $\Omega$ be arbitrary set. Let $Q$ be a partition of $\Omega$. I already proved that the collection of all unions of the cells in $Q$ is a complete field $\mathcal{F}$ (complete field is ...
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2answers
15 views

limit points of subset of real numbers

Let $$A=\{ \frac{\sqrt{m} -\sqrt{n}}{\sqrt{m}+\sqrt{n}} | m,n\in \Bbb{N} \}$$ I think that we must find sequence of $A$ and find limit of sequence,let $a_m =\frac{\sqrt{ k ^2 m^2} -\sqrt{ ...
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2answers
38 views

Finding limit points for these sets

Here's my resoning for finding limit points for some sets. Could you guys read it and see if it's all good? <3 $$\{(x,y)\mid \ x^2+y^2<1\}$$ For this set, its kinda simple to see that every ...
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3answers
46 views

Discontinuous maps taking compacts to compacts

It's commonly known that in general topology, a continuous map $f$ from a topological space $(X, \tau)$ to another topological space $(Y, \tau')$ will send every compact set to another compact set. ...
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46 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
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0answers
36 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
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0answers
34 views

Topology problem [duplicate]

It is prob.5(a) of section 16, ch2 in munkres: Let X and X' denote a single set in the topologies T and T',respectively. Let Y and Y' denote a single set in the topologies U and U', respectively. ...
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6answers
61 views

Prove that $\{(x,y)\mid xy>0\}$ is open

I need to prove this using open balls. So the general idea is to construct a open ball around a point of the set. A point $(x,y)$ such that $xy>0$. Then we must prove that this ball is inside the ...
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1answer
25 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 Let $A$ and $B$ be LCA-groups and $H$ a (not ...
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2answers
60 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
26 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
42 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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0answers
31 views

What is the definition of the following concepts and how I can characterize each concept. [on hold]

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
33 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
39 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 ...
2
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1answer
19 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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0answers
29 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 ...
2
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2answers
27 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
55 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 ...
3
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1answer
49 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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0answers
44 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
26 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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0answers
52 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
69 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 ...
2
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1answer
39 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
51 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
35 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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0answers
30 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
44 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 ...
3
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1answer
64 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
39 views

Connected set imply continuous boundary

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
35 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
40 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
68 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 ...
2
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3answers
72 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 ...
2
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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
144 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
28 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
15 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
52 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
200 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 ...