Basic Analysis I — Real Analysis · Problem 7.2.10

Metric Spaces — Jiri Lebl

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

Problem

Suppose

\{S_i\}

i \in \mathbb{N}

, is a collection of connected subsets of a metric space

(X, d)

, and there exists an

x \in X

such that

x \in S_i

for all

i \in \mathbb{N}

. Show that

\bigcup_{i=1}^\infty S_i

is connected.