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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29 views

Proof of the Harnack inequality

Let $\Omega\subseteq\mathbb{R}^n$ be a domain, $\Omega'\subset\subset\Omega$ be a domain and $u\in C^0(\overline{\Omega})$. Suppose we know $$\sup_{\Omega'}u\le 3^n\inf_{\Omega'}u\tag{1}$$ if ...
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0answers
37 views

Topologically equivalent metrics, using different definitions.

I´ve been dealing with topologically equivalent metrics for a while, using the usual definition, that $d$ and $d'$ are topologically equivalent iff they have the same open sets. However, there is ...
-2
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0answers
37 views

Proofs involving 3 quantifiers: A(3,3)=6 cases [closed]

I knows how to prove statement involving 1 or 2 quantifiers. So there are 6 combinations of 3 universal quantifiers ("for all" and "there exist") with an extra implication that makes a quantified ...
0
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1answer
66 views

Prerequisites for learning general topology

I want to learn general topology in order to apply it in electromagnetism. I am an undergraduate student and I have a background in linear algebra (not at an advanced level), linear differential ...
2
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1answer
24 views

What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other?

Let $\tau ,\tau_1$ be two topologies on the set $\mathbb R$ .Suppose $\tau \subset \tau_1$ .What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other? ...
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2answers
97 views

Is the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?

I know the closure of $\mathbb Q$ is $\mathbb R$, but does this imply that the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?
3
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1answer
36 views

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent:

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent: a) $f$ is uniformly continuous in $X$. b)For every pair of sequences $(x_n), (y_n) \subseteq X$ such that $ d(x_n, ...
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1answer
65 views

Is Cantor set closed? [closed]

https://www.youtube.com/watch?v=dazO9UoKmyA This professor said Cantor set is closed because it's FINITE union of closed intervals at 14.00. But isn't it a wrong statement since Cantor set is ...
4
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1answer
68 views

Does a space with peoperty A have a topological name?

As we know, If $X$ is a Tychonoff pseudocompact space, then for every decreasing sequence $\cdots\subset W_2\subset W_1$ of nonempty open subsets of $X$ the intersection $\bigcap_{i=1}^{\infty} ...
3
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1answer
36 views

Show that $A=\bigcap G_{A}$

Given a metric space $(X,d)$ and $A\subset X$, let $G_{A}$ be the set which consists of all the open sets that contain $A$. Show that $A=\bigcap_{B \in G_{A}}B$ It is obvious that $A \subset ...
7
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2answers
104 views

Hölder continuous functions are of 1st category in $C[0,1]$

I'm trying to show that the Hölder continuous functions in $[0,1]$ are a set of first category in $C[0,1]$. Does it suffice to show that they are not an open subset of $C[0,1]$? Let ...
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1answer
15 views

Covers of X in the Stone-Cech compacticatoin

Suppose that $X$ is a Hausdorff completely regular space. $X$ embeds homeomorphically into its Stone-Cech compactifaction $\beta{X}$ and is dense in $\beta{X}$; we can identify $X \subset ...
3
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1answer
35 views

Any open subset of $\mathbb R$ is an uniquely countable union of disjoint open intervals

I know how to prove that any open subset of $\mathbb R$ is a countable union of disjoint open intervals. (see this question: Any open subset of $\Bbb R$ is a at most countable union of disjoint open ...
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3answers
120 views

Can we fit uncountably many nonempty open sets in $\mathbb{R}^n$ such that each point is contained in at most finitely many of them?

This simple question came to my mind the other day: Question: Can we fit uncountably many nonempty open sets in $\mathbb{R}^n$ such that each point of $\mathbb{R}^n$ is contained in at most ...
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0answers
14 views

X a locally compact Haussdorf space, each singleton of which is the intersection of countably many open sets. Show that X is first countable. [duplicate]

Let $X$ be a locally compact Haussdorf space, each singleton of which is the intersection of countably many open sets. Show that X is first countable.
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2answers
32 views

Inverse mapping on a set $U_1\times U_2$, wrong intuition?

Let $f(x) = (f_1(x),f_2(x))$ where $f: X\to Y_1\times Y_2$. And $f_1:X\to Y_1, f_2: X\to Y_2$ where $X,Y_1,Y_2$ are topological spaces. I want to prove some continuity properties, but my ...
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0answers
20 views

Topological Embedding Which is Neither Open nor Closed

I'm having trouble coming up with an example of an embedding which is neither open nor closed. My attempts have included trying to find such a map from $\mathbb{R}$ (given the usual Euclidean ...
6
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1answer
46 views

Is every element contained in a smallest measurable set?

Let $(X,\mathcal F)$ be a measure space, then for each $x \in X$ does there always exists a smallest measurable set containing $x$? If $X$ is countable or $\mathcal F$ finite, then this is true, as ...
1
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1answer
40 views

Continuity of a product of two real valued continuous function.

In the basic analysis, we proved the following with the $\epsilon - \delta$ method. For metric spaces $X$ and $\mathbb{R}$, $fg:X\rightarrow \mathbb{R}$ is continuous, if $f,g:X\rightarrow ...
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1answer
53 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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1answer
20 views

Countable Product of Sequentially Compact spaces is Sequentially Compact

I would like to prove that a countable product $$\prod_{i \in \mathbb{N}}X_i $$ of sequentially compact spaces $X_i$ is sequentially compact. That is, any sequence in the product space has a ...
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1answer
9 views

$B(\tau)\bigcap Y=B(\tau_{Y})$?

$(X,\tau)$ is a topological space, $Y$ is a subset of $X$. $B(X)=B(\tau)$ is Borel $\sigma-$algebra on $X$. There are two ways to generate $\sigma-$algebra on $Y$. First, $(Y,\tau_{Y})$ is a ...
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0answers
17 views

The set of limit points in a $T_1$ space is closed [duplicate]

Prove the set of limit points in a $T_1$ space is closed. Definitions: Limit Point: $x$ is a limit point of a set if every open set that contains $x$ contains at least one other point of the ...
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0answers
24 views

Stone-Cech compactification of a dense subset of compact Hausdorff space

Let $X$ be a dense subset of compact Hausdorff space Y. Every continous function $f:X \to [0,1]$ extends to a continous function $\bar{f} : Y\to [0,1]$ ($\bar{f}(x) = f(x) \; for \; x \in X$). I want ...
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2answers
18 views

countable product of totally bounded space is totally bounded

Let $ \{ X_i \}_ { i \in \mathbb{N} }$ be a countable collection of metric spaces $(X_i, d_i)$. The product topology on product space $X=\prod_{i=1}^{\infty} X_i$ is equivalent to the metric topology ...
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0answers
32 views

A confusion of a real analysis online lecture: Relative compactness

https://www.youtube.com/watch?v=kkKfRaI-cqs At 13.00 what does the professor mean to let those subcovers be "restricted" to Y? Is that a process like A is contained by B implies that A intersect C in ...
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0answers
15 views

core-compact but not locally copact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
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1answer
61 views

Linear application on a normed space

How to prove that if $E$ and $F$ are tow normed spaces with $dim(F)<\infty$ and $f\in L(E,F)$ Then $$f~ \text{open} \Longleftrightarrow f ~\text{surjective}$$ If i suppose that $f$ is open, then ...
2
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1answer
26 views

Topological field - Proving continuity of inversion

Given a field $F$ and an absolute value $|\ |$ on $F$, define the distance $d(x,y)$ between two elements $x,y\in F$ by $$ d(x,y) = |x - y|. $$ I just worked through the proofs that $d$ defines a ...
2
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1answer
52 views

The maximality of a family of pairwise disjoint meager open sets implies the denseness of its union

Consider the following theorem from A. Kechris's Classical Descriptive Set Theory: (8.29) Theorem. Let $X$ be a topological space and $A \subseteq X$. Put $$U(A) = \bigcup \{ U\text{ open} : U ...
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2answers
33 views

continuous image of a locally compact space is locally compact

Is continuous image of a locally compact space is locally compact? Let $X$ be locally compact(l.c.).Let $f:X\to Y$ is continuous and surjective. A space $X$ is locally compact if for each $x\in X$ ...
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1answer
36 views

Is this bullet really needed in Furstenberg's proof of infinitude of primes?

See here . The bullet I'm referring to is: Any union of open sets is open: for any collection of open sets $U_i$ and $x$ in their union $U$, any of the numbers $a_i$ for which $S(a_i, x) \subset ...
1
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1answer
31 views

The intersection of a locally finite family of open sets is a closed set?

Let $\{ U_i : i \in I \}$ be a locally finite family of open sets, if $I$ is infinite index, and there exist infinite different $U_i$, must $\bigcap_{i \in I} U_i $ be a closed set? Or, further more, ...
4
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1answer
28 views

Is $L^1_{loc}(\mathbb{R})$ complete with the norm $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy$

Let $BL^1_{loc}$ be the space of locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy<\infty$. Is this space complete ? What I tried: ...
4
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0answers
27 views

Compactness and directed systems of subspaces

Let $X$ be a topological space and let $K$ be a subspace of $X$. It is easy to verify the claim below: Let $\{ U_j : j \in J \}$ be a directed system of open subspaces of $X$ with the following ...
0
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0answers
7 views

Line integrals in a double connected set

If P and Q are continuously differentiable on an open doubly connected(one hole) region $R$, and if $\partial P/\partial y = \partial Q/\partial x$ everywhere in $R$, how many distinct values are ...
1
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2answers
60 views

Topology-book for self study about physical meaning

I self study physics, and I have come across some mathematics that are unknown to me. It has to do with helicity, linking numbers and other topological factors. A friend of mine told me that this is ...
6
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1answer
89 views

Characterization of open maps in terms of nets

Here I asked about characterization of closed maps in terms of nets/sequences. I find this view illuminating, so I wanted to ask about open maps. A map $f: X \to Y$ is open if for each $x \in X$ and ...
1
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2answers
37 views

How to show that the right half plane of $\mathbb{R}^2$ is homeomorphic to the open unit disk?

I am trying to show that $\{(x,y)\in\mathbb{R}^2:x>0\}$ is homeomorphic to the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$. I know that the disk is homeomorphic to the whole plane. A homeomorphism ...
1
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1answer
20 views

Discrete metric, countable basis?

Give an example of a metric space which does not have a countable basis. I was thinking of some uncountable set, with a metric which results in an uncountable number of open subsets. Which ...
2
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0answers
62 views

Union of infinite broom and topologist's sine, connectednes, locally connectednes properties…

I'd like to know if my answer of the following exercise is correct. I really appreciate any suggestion you can provide to improve my argument or corrections in case I made a mistake :) Let ...
2
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1answer
28 views

Where do I use the fact that $V$ is closed in $X$ in the following proof.

Let $V$ be a supspace of $(X,\tau)$ and $A\subseteq V$. Let $V$ be closed in $X$. Then $A$ is closed in $V$ if and only if $A$ is closed in $X$. Where the subspace $V$ has the relative ...
2
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1answer
52 views

Examples of arguments from connectedness

Suppose $X$ is a connected topological space. A typical way that we prove a property $P(x)$ holds for all $x \in X$ is to show that $P$ is an open and a closed condition, and that $P(x)$ for some $x ...
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1answer
25 views

Tietze extension into $ [0,1)$

Prove or disprove that a continuous real valued function on a closed subset of a normal space into $[0,1)$ can be extended to a continuous real valued function on the entire space into $[0,1)$. I ...
2
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1answer
23 views

Is AC necessary to show that in metric spaces $x\in\operatorname{closure}(A)$ implies $\exists\{a_n\}_{n=1}^\infty\subseteq A$ s.t. $\lim a_n=x$?

Let $(X,d)$ be a metric space. Let $x\in\operatorname{closure}(A)$, where $A\subseteq X$. Then for each $n\in\mathbb{N},\exists x_n\in B_{\frac{1}{n}}(x)\cap A$, where $B_\varepsilon(x)$ is the open ...
2
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0answers
47 views

How much a simply connected region can be wonderful? [closed]

How much a simply connected region in euclidean spaces(specially $\mathbb{R}^2$) can be wonderful? As an example I have the below example (take the inner region of the below boundary as the space) in ...
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5answers
63 views

Show that $f(x) = 0$ for all $x \in \mathbb{R}$ [duplicate]

$f: \mathbb{R} \to \mathbb{R}$ is continuous with $f(x)=0$ for all $x \in \mathbb{Q}$. Show that $f(x) = 0$ for all $x \in \mathbb{R}$. Can anyone please point me in the right direction as to how ...
1
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2answers
43 views

compactness of Hilbert cube

I want to show that the Hilbert cube which is: $H=\{(x_1,x_2,...) \in [0,1]^{\infty} : for \ each \ n \in \mathbb{N}, |x_n|\leq \dfrac{1}{2^n}\}$ is compact with respect to the metric: ...
3
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2answers
44 views

Any quotient of a compactly generated space is compactly generated

I found a note about compactly generated. This is the article http://www.math.uiuc.edu/~franklan/Math535_1205.pdf. I worry whether the proof of Proposition 2.4 is true. I not understand why the ...
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2answers
18 views

On Closure of Product subset of $\Bbb R×\Bbb R$

Suppose that $\Bbb R×\Bbb R$ has the standard topology. If $S=\left\{(t,\sin{\frac{1}{t}})\mid t\in R\text{ and }t\gt 0\right\}$. Show that $(0,0)$ $\in \overline{S} $