Intersections of Open Sets and Unions of Closed Sets

The sets ${\mathcal {G}}^{\delta }$ and ${\mathcal {F}}^{\sigma }$ are a subset of the Borel set. The set ${\mathcal {G}}^{\delta }$ is the collection of all sets which are a countable intersections of open sets and the set ${\mathcal {F}}^{\sigma }$ is the collection of all sets which are a countable union of closed sets.

Definitions

Let 'X' be a topological space whose collection of open sets is denoted '<math>\tau<\math>' and whose collection of closed sets is denoted 'C', then,

<math>\mathcal{G}^\delta = \{\bigcap_{n=1}^\infty\mathscr{O}_n: \mathscr{O}_n\in\tau \forall n\in\mathbb{N}\}<\math>

<math>\mathcal{F}^\sigma = \{\bigcup_{n=1}^\infty C_n: C_n\in C \forall n\in\mathbb{N}\}<\math>

The definitions can be extended as follows. Let 'w' be a non-trivial word in the alphabet <math>\{\delta,\sigma\}<\math> of length 'm'. Let 'u' be the first 'm-1' letters in the word and let <math>w_m<\math> be the last letter. Then we define,

<math>\mathcal{F}^w = \mathcal{G}^w = \{\bigcap_{n=1}^\infty P_n: P_n\in\G^u \forall n\in\mathbb{N}\}<\math>

if <math>w_m = \delta<\math> and we define

<math>\mathcal{F}^w = \mathcal{G}^w = \{\bigcup_{n=1}^\infty P_n: P_n\in\G^u \forall n\in\mathbb{N}\}<\math>

if <math>w_m = \sigma<\math>