- #1

madness

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## Homework Statement

Let A, B be closed non-empty subsets of a topological space X with [tex] A \cup B [/tex] and [tex] A \cap B [/tex] connected.

Prove that A and B are connected.

## Homework Equations

A set Q is not connected (disconnected) if it is expressible as a disjoint union of open sets, [tex] Q = S \cup T [/tex]

## The Attempt at a Solution

I'm trying a proof by contradiction.

By the above definition, a set which is not connected must be open (is this really true?). So start by assuming A is disconnected, ie [tex] A = C \cup D [/tex] for C, D open and disjoint. Then A must be open, but should also be closed. Now consider [tex] A \cup B = C \cup D \cup B [/tex] and [tex] A \cap B = C \cup D \cap B [/tex]. I want to to arrive at a contradiction. The given properties are that A is both open and closed, B is closed, C and D are open and disjoint and [tex] A \cup B = C \cup D \cup B [/tex] and [tex] A \cap B = C \cup D \cap B [/tex] are connected. This all seems very complicated.