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Topology

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Moderate

Application Of Compactness

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Which of the following statement(s) is(are) TRUE?

A

A simplicial graph of $n$ edges is $n$ of copies of interval $[0,1]$ with some of their end points identified. A loop on a simplicial graph with infinitely many edges only passes through the midpoints of finitely many edges.

B

$f$ is a continuous function on a compact space $X$. $\forall x,y\in X$, let $F(x,y)=f(x)+f(y)-f(x)f(y)$, then $F$ has maximum.

C

The union of a sequence of compact subsets in a topological space is compact.