Section 23: Problem 3 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 $\{A_{\alpha}\}$ be a collection of connected subspaces of $X$ ; let $A$ be a connected subspace of $X$ . Show that if $A\cap A_{\alpha}\neq\emptyset$ for all $\alpha$ , then $A\cup(\cup A\alpha)$ is connected.

If there is a separation $(U,V)$ of the union, then $A$ and all $A_{\alpha}$ lie within $U$ or $V$ (Lemma 23.2). Suppose, $A\subset U$ , then each $A_{\alpha}\subset U$ , and $V$ is empty. Contradiction.

Alternatively, the connected (by Theorem 23.3) subspaces $\{A\cup A_{\alpha}\}$ share a point (apply Theorem 23.3 again).

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Section 23: Problem 3 Solution

Section 23: Problem 3 Solution