Suppose a set M is closed and bounded. Prove that M has a maximal and minimal element.

6. Let X be a topological space, and let A be a subset of X which is both open and closed. Show that A is a union of connected components of X.

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

Let {V₁,..., Vn} be a spanning set for an inner product space V. Suppose that x and y are such that (x, vk) = (y, vk) for all k. Prove that x = y. . Please prove this from definitions and do not refer to previous review, hw, or gw problems (you will lose points if you do so). If you use a theorem or property from the text (that has been proved), please refer to which one.

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