Tarea 10, Análisis real

Due October 16

Problem 1

If $latex X$ is discrete, then $latex (\mathcal C_X,d_H)$ is discrete.

Problem 2

Let $latex A\subset X$ be a finite set of isolated points in $latex X$. Then $latex A$ is isolated in $latex \mathcal C_X$.

Problem 3

Let $latex A_n$ be a decreasing sequence of nonempty compact sets in $latex X$. Then

$latex \displaystyle \lim A_n = \bigcap_{n\ge 1} A_n$

in $latex \mathcal C_H$.

Problem 4

Let $latex f_1, \ldots, f_N:X\to X$ be contractions in the complete metric space $latex X$, and $latex K$ the self-similar set with respect to the $latex f_i$. If $latex A\subset X$ is compact and

$latex A\subset f_1(A)\cup\ldots\cup f_N(A)$,

then $latex A\subset K$.

Problem 5

Under the same hypothesis of the previous problem, if $latex B\subset X$ is nonempty and

$latex B\supset f_1(B)\cup\ldots\cup f_N(B)$,

then $latex \bar B \supset K$.