Show that a union of any finite points of bounded subsets of a metric space is bounded

Let A be a closed bounded subset of a metric space (X, d). Is A acompact subset? Explain why.

let fiR> R the function defined in the fllowig farm fox=x²,indicate whether itisa continuous function Bases of Topology R which imeludes the family of all open sets using

7. Let X be a locally path-connected space. Let U be an open, connected subset of X. Show that U is path-connected.

A metric space is said to be separable if it contains a countable dense subset. (a) Is R separable? (b) Is Rn separable?

Differentiate between Euclidean and complex Euclidean spaces. How Schwartz inequality is modified in the case of complex Euclidean spaces?