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COMPACT SPACE

Compact의 정의는 다음과 같다.

Definition
A space X is said to be compact if every open covering $\mathcal{A}$ of $X$ contains a finite subcollection that also covers.

다시 말해서 어떤 공간 $X$를 감싸는 open set들의 모임(open covering $\mathcal{A}$)이 있다면 그 중 임의로 유한개의 open set들을 뽑아서 다시 $X$를 감쌀 수 있을 때(또는 망라할 수 있을때) $X$는 compact이다.

Lemma
Let $Y$ be a subspace of $X$. Then $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$.

Proof
$(\iff)$
If $Y$ is compact,then there must exist a finite subcollection $\mathcal{A}^{\prime}$ of open covering $Y$,defined by $\lbrace A^{\prime}_ {i} \rbrace_{i \in I}$ ,where $I$ is a finite index set, and this set is equivalent to $ \lbrace A^{\prime}_ {i} = A_{i} \cap Y \mid A_ {i} \subseteq_{open} X \rbrace_{i \in I}$, where $\lbrace A_{i}\rbrace_{i \in I}$ is a finite subcollection of $X$.
This is a highlight test.

Normal block

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alert('Hello World!');

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print 'helloworld'

Highlight block

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alert( 'Hello, world!' );

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print 'helloworld'

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def foo
  puts 'foo'
end

def foo
  puts 'foo'
end
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def foo
  puts 'foo'
end

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#include <iostream>

using namespace std;

void foo(int arg1, int arg2)
{

}

int main()
{
  string str;
  foo(1, 2);
  cout << "Hello World" << endl;
  return 0;
}
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Posted on | In

One of the rewards of switching my website to Jekyll is the ability to support MathJax, which means I can write LaTeX-like equations that get nicely displayed in a web browser, like this one \( \sqrt{\frac{n!}{k!(n-k)!}} \) or this one \( x^2 + y^2 = r^2 \).

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