Properties of functions on topological spaces

by vidyarthi   Last Updated January 16, 2018 09:20 AM

Three questions follow which are related to concept of topological spaces and functions defined thereon.

a)Let $F$ be a family of functions on a compact topological space $X$ such that it is closed under multiplication of functions and for every $x\in X\exists$ an open neighbourhood $U$ such that the restriction of a function $f\in F$ is identically zero. Then, does the zero function belong to $F$?

b)Is the space of Lipschitz continuous functions on $[0,1]$ with lipschitz constant $k=1$ compact under the sup-norm?

c)If a family of functions belonging to the set of continuous functions on $[0,1]$ has the property that every subfamily vanish a certain point, then theredoes there exist a point which is a common zero for all members of the family?

Am totally clueless on the three problems, though I think all are true. May be, for a), the existence of a finite subcover for every open cover has a role to play. As for b), I have a feeling that it is false since the limit of a sequence of lipschitz functions need not be lipschitz. Again, for c), since the family of functions need not be equicontinuous, therefore the conclusion may be false. Any hints. Thanks beforehand.

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