Tarea 10, Análisis real

Due October 16

Problem 1

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

Problem 2

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

Problem 3

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

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

in $\mathcal C_H$.

Problem 4

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

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

then $A\subset K$.

Problem 5

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

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

then $\bar B \supset K$.