Real Analysis II (2024-2025)

A subset A of complete metric space M is complete iff A is ________

• open

• closed

• bounded

• dense in M

A closed set A in a metric space M is nowhere dense if and only if complement of A is_____

• open

• nowhere dense

• dense

• closed

Which of the following spaces are not complete?

• R with usual metric is complete

• Q with usual metric is complete

• Any discrete metric space is complete

True or False: Any subset of a discrete metric space has no limit point.

• True

• False

True or False: The set Q of rational number is not closed in R.

• True

• False

Diameter of [0,1] is_______

• 0

• 1

• 2

• 3

In R with usual metric, then find B ( -1, 1) ________

• (2,0)

• (-1, 0)

• (-2,0)

• (0,2)

The diameter of any non-empty subset in a discrete metric space is _____

• 0

• 1

• 2

• 3

R is of ________

• first category

• second category

True or False: any cauchy sequence is convergent sequence.

• true

• false