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.

Related Questions

Question on Properties of continuous function

Updated May 14, 2017 04:20 AM