Basic Analysis I — Real Analysis · Problem 7.2.17

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · open-and-closed-sets

Problem

Let

(X, d)

be a metric space. a) Prove that for every

x \in X

, there either exists a

\delta > 0

such that

B(x,\delta) = \{x\}

, or

B(x,\delta)

is infinite for every

\delta > 0

. b) Find an explicit example of

(X, d)

X

infinite, where for every

\delta > 0

and every

x \in X

, the ball

B(x,\delta)

is finite. c) Find an explicit example of

(X, d)

where for every

\delta > 0

and every

x \in X

, the ball

B(x,\delta)

is countably infinite. d) Prove that if

X

is uncountable, then there exists an

x \in X

and a

\delta > 0

such that

B(x,\delta)

is uncountable.