Section 23: Problem 2 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $\{A_{n}\}$ be a sequence of connected subspaces of $X$ , such that $A_{n}\cap A_{n+1}\neq\emptyset$ for all $n$ . Show that $\cup A_{n}$ is connected.

If $\cup A_{n}$ is disconnected then there is a separation $(U,V)$ of the union, then each set of the sequence being connected lies within either $U$ or $V$ (Lemma 23.2). Suppose that $A_{1}\subset U$ , then, by induction, each $A_{n}\subset U$ , and $V$ is empty. Contradiction.

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Section 23: Problem 2 Solution

Section 23: Problem 2 Solution