In what condition the set of all isolated point in a 2nd countable metric space is empty?
$NOTE :-$ Sir Brian Scott's answer is ok , but I would like an answer of this edited form , Thanks in advance
Asked
Active
Viewed 91 times
1 Answers
3
I take it that you’re asking how big the Cantor-Bendixson rank of a second countable metric space can be. Every countable ordinal is a second countable, metrizable space, and there are countable limit ordinals of arbitrarily large countable rank, so
$$\sup\{\operatorname{rank}(X):X\text{ is second countable and metrizable}\}\ge\omega_1\;,$$
where $\operatorname{rank}(X)$ denotes the Cantor-Bendixson ranks of $X$. On the other hand, if $\operatorname{rank}(X)>\omega_1$, then there is a sequence $\langle x_\xi:\xi<\omega_1\rangle$ of distinct points of $X$ such that if $H=\{x_\xi:\xi<\omega_1\}$, then $\{x_\xi:\xi<\eta\}$ is open in $H$ for each $\eta<\omega_1$. Clearly $H$ is not Lindelöf, so $X$ cannot be second countable. Thus,
$$\sup\{\operatorname{rank}(X):X\text{ is second countable and metrizable}\}=\omega_1\;.$$
Brian M. Scott
- 616,228