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Tarea 12, Análisis real

Due October 30

Problem 1

Let $latex A\subset X$ be connected. Then $latex \bar A$ is connected.

Problem 2

If $latex A\subset X$ is connected and $latex A\subset B\subset \bar A$, then $latex B$ is connected.

Problem 3

$latex X$ is connected if and only if every continuous $latex f:X\to Y$ into a discrete space $latex Y$ is constant.

Problem 4

If $latex A$ is convex, then $latex \bar A$ is convex.

Problem 5

Prove $latex B_r(x_0)$ is convex from the fact that $latex B_1(0)$ is convex.