Luiz Cordeiro
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1 answers
25 votes
13k views
How to show the that a set $A$ nowhere dense is equivalent to the complement of $A$ containing a dense open set?
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32 votes

First, you should know that, for any $B\subseteq X$, $X\setminus\overline{B}=(X\setminus B)^\circ$ and that $X\setminus B^\circ=\overline{X\setminus B}$. Now \begin{align*} A\text{ nowhere dense }&...

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4 answers
11 votes
8k views
Showing that the closure of a totally bounded set is totally bounded
27 votes

Just as you mentioned, given $\varepsilon>0$, we can find $x_1,\ldots,x_n\in M$ such that $M\subseteq \bigcup_i B_{\varepsilon/2}(x_i)$. Let's show that $\operatorname{cl}(M)\subseteq\bigcup_i B_\...

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6 answers
24 votes
9k views
What is the sum of all positive even divisors of 1000?
23 votes

Since $1000=2^3\cdot5^3$, the even divisors of $1000$ have the form $2^i5^j$, where $1\leq i\leq 3$ and $0\leq j\leq 3$. There are only 12 of them, so you can do this calculation directly. ...

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4 answers
23 votes
25k views
The closure of a connected set in a topological space is connected
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20 votes

Suppose that $E$ is connected. Let $A,B\subseteq X$ be separated sets (that is, $\overline{A}\cap B=A\cap\overline{B}=\varnothing$) such that $\overline{E}=A\cup B$, and suppose that $A\neq\varnothing$...

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12 answers
34 votes
2k views
Why aren't vacuous truths just undefined?
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19 votes

This is done so that classical propositional calculus follows some natural rules. Let's try to motivate this, without getting into technical details: The expression "$P\Rightarrow Q$" should be read "...

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6 answers
1 votes
1k views
If $A^2$ is the zero matrix, show that $A$ is linearly dependent?
13 votes

As you said, $A$ is a square matrix. Since $A^2=0$, then $$0=\det 0=\det(A^2)=\det(A)^2$$ so $\det A=0$, and this means that the rows and columns of $A$ are LD.

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1 answers
11 votes
3k views
A function with countable discontinuities is Borel measurable.
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13 votes

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function with countable discontinuities, and let $c\in\mathbb{R}$. We must prove that $f^{-1}(-\infty,c)$ is measurable. Let $A=\text{int}f^{-1}(-\infty,c)$ (...

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4 answers
18 votes
2k views
Are professional mathematicians concerned with formalizing infinitely many dependent choices?
12 votes

This is basically a long comment: The thing is that when we construct sequences in the manner you described ("Having picked $x_1,x_2,\ldots,x_n\in S$, pick $x_{k+1}$,...") is much simpler to grasp ...

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3 answers
12 votes
1k views
Why does metric space which has the countable chain condition implies separable?
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12 votes

Let $X$ be a (non-empty) c.c.c. metric space. Let us temporarily fix a positive integer $n\geq 1$. Let $\mathscr{F}_n$ be the family of classes of open balls of radius $1/n$ (in $X$) which are ...

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1 answers
9 votes
2k views
Compactness and sequential compactness in metric spaces
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10 votes

Theorem Let $(M,d)$ be a metric space. The following are equivalent: (a) $M$ is compact; (b) $M$ is sequentially compact; (c) $M$ is complete and totally bounded. Proof: (a$\Rightarrow$b) ...

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2 answers
16 votes
2k views
Does the graph of a measurable function always have zero measure?
10 votes

No. First, recall that the product measure is not unique for non-$\sigma$-finite spaces. Let $(X,\mathscr{M},\mu)=(\mathbb{R},2^\mathbb{R},\#)$ and $f(x)=|x|$. We consider the product measure $$\nu(A)...

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4 answers
20 votes
18k views
Inverse image of a compact set is compact
10 votes

This is not true in general. Let $X=Y=[0,1]$. Take $X$ with the usual topology. For $Y$, take the topology $$\tau=\left\{\varnothing,Y,(1/2,1]\right\}.$$ Then $id:x\in X\mapsto x\in Y$ is continuous, ...

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1 answers
5 votes
2k views
What are the range and the norm of this bounded linear operator?
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10 votes

It is obvious that $\Vert Tx\Vert_\infty\leq\Vert x\Vert_\infty$, hence $\Vert T\Vert\leq 1$. Since $\Vert T(1,0,0,\ldots)\Vert_\infty=\Vert (1,0,\ldots)\Vert_\infty$, we conclude that $\Vert T\Vert=1$...

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5 answers
6 votes
2k views
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
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9 votes

The exponential funtion $t\mapsto \exp(t)$ is convex, so $$\begin{align} xy&=\exp(\log(xy))\\ &=\exp(\log(x)+\log(y))\\ &=\exp((1/p)\log(x^p)+(1/q)\log(y^q))\\ &\leq (1/p)\exp(\log(x^p)...

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2 answers
7 votes
1k views
Compact metric connected space
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9 votes

Suppose $X$ wasn't connected. Let $A$ be a non-trivial clopen in $X$, and let $B=X\setminus A$. Then $A$ and $B$ are closed, hence compact, subsets of $X$, so there exist $a\in A$ and $b\in B$ such ...

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4 answers
7 votes
2k views
How to prove that the set $A = \left\{ {p:{p^2} < 2,p \in {\Bbb Q^+}} \right\}$ has no greatest element?
9 votes

To show that $A$ has no greatest element, you have to show that there is no $q\in A$ which is greater than any $p\in A$, the opposite of your first line. If $p^2<2$, we take the number $q=p+\frac{...

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1 answers
1 votes
822 views
Why are the isomorphisms and bijective morphisms not identical in the category of Pos?
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9 votes

Take two elements, $0$ and $1$, and define the poset $X$ with $0\leq 1$ (and the reflexivity conditions), and $Y$ the poset only with $0\leq 0$ and $1\leq 1$ (i.e., we can't compare $0$ and $1$ in $Y$....

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1 answers
4 votes
2k views
Prove that an infinite matrix defines a compact operator on $l^2$.
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8 votes

One of the main tools on infinite matrices and Hilbert spaces operators is the so-called Schur's test. This is Exercise 45 in Halmos' A Hilbert Space Problem Book. Schur's test. Let $A=[a_{ij}]_{i,...

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2 answers
4 votes
85 views
$G$ is finite abelian group, do we have $\{ng : g\in G \} = G$ for $(n, |G|) = 1 $ (coprime)?
7 votes

You proof is correct and looks like the easiest one. Here's one without using the fundamental theorem of fg abelian groups, but on the same spirit: By Bézout's theorem, take $k,l\in\mathbb{Z}$ with $...

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3 answers
4 votes
120 views
Proving that $a^{25} \bmod 65 = a \bmod 65$?
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7 votes

If $a=0$ then this is trivial, so assume $a\neq 0$. $\mathbb{Z}_5=\mathbb{Z}/5\mathbb{Z}$ is a field, so $\mathbb{Z}_5^\times$, the group of invertible elements, is a group of $4$ elements. In ...

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3 answers
25 votes
16k views
The dual space of $c$ is $\ell^1$
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7 votes

The last argument where you say that $f(x)$ would be unbounded does not seem valid since you only have $f(x)\leq \Vert x\Vert_\infty\sum_{k=1}^\infty|f(e_k)|$. If $\sum_{k=1}^\infty |f(e_k)|=\infty$, ...

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2 answers
7 votes
947 views
Can the associative property of a group be followed from its other properties?
7 votes

No. Let $A=\left\{0,1,2\right\}$ with the operation $+:A\times A\rightarrow A$ given by $0+0=0$, $0+1=1$, $0+2=2$, $1+0=1$, $1+1=0$, $1+2=1$, $2+0=2$, $2+1=2$, $2+2=0$. Then $0$ is the (bilateral) ...

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2 answers
3 votes
436 views
If $\operatorname{rank}(A)=m$, can we say anything about $\operatorname{rank}(AA^t)$?
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7 votes

It's true in general that $rank(A^tA)=rank(A)$. To prove that, it clearly suffices to show that $kerA^tA=ker A$, since $rank(T)=n-dim(ker T)$ for every $n\times n$ matrix $T$ (where we denote $ker T=\...

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3 answers
3 votes
768 views
Question on integral, notation and Nikodym derivative
6 votes

Remember from one-variable calculus that, given a (continuous) function $f:\mathbb{R}\to\mathbb{R}$, the indefinite integral of $f$ is the integral of $f$ (with respect to Lebesgue measure) without a ...

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4 answers
4 votes
85 views
Solving $ \{ \varnothing, \{ x\} \} = \{ y, \{ \varnothing \} \}$.
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6 votes

Well, you have to look at several cases: Since $\varnothing$ is a member of the left-hand side, then it is also a member of the right-hand side. Since the members of the right-hand side are $y$ and $\...

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4 answers
1 votes
42 views
Set of $n$ numbers which has no two elements whose sum is an another
6 votes

Take any set of odd numbers. $\hspace{0pt}$

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8 answers
3 votes
4k views
What is an example of a groupoid which is not a semigroup?
6 votes

These are called magmas, not groupoids. The ``midpoint'' operation $s\ast t=\frac{s+t}{2}$ on $\mathbb{R}$ makes it a magma which is not a semigroup.

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6 answers
15 votes
12k views
If the diameter of graph is greater than 3 then the diameter of its complement graph is less than 3
6 votes

Denote by $d_G$ and $d_{G'}$ the distances in $G$ and $G'$, respectively. Take vertices $u,v$. There are a few cases of interest. If $d_G(u,v)\geq 2$, then the edge $uv$ is not in $G$, so $uv\in G'$ ...

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2 answers
7 votes
2k views
Chain Rule and Vector valued functions?
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6 votes

Let's recap some things: For general $f=(f_1,\ldots,f_m):U\subseteq\mathbb{R}^n\to\mathbb{R}^m$ (where $U$, the domain of $f$, is open in $\mathbb{R}^n$) and $x\in U$, $Df(x)$ is the Jacobian (or ...

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1 answers
4 votes
772 views
Question on one point compactification
Accepted answer
6 votes

Since $X\in\tau\subseteq\tau^*$, $X$ is open in $X^*$. For density, the only point in $X^*\setminus X$ is $\infty$. Let $V$ be an open neighbourhood (in $X^*$) of $\infty$. Then $X\setminus V$ is ...

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