Description: The finite complement topology on a set A . Example 3 in Munkres p. 77. (This version of fctop requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | fctop2 | |- ( A e. V -> { x e. ~P A | ( ( A \ x ) ~< _om \/ x = (/) ) } e. ( TopOn ` A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isfinite | |- ( ( A \ x ) e. Fin <-> ( A \ x ) ~< _om ) |
|
| 2 | 1 | orbi1i | |- ( ( ( A \ x ) e. Fin \/ x = (/) ) <-> ( ( A \ x ) ~< _om \/ x = (/) ) ) |
| 3 | 2 | rabbii | |- { x e. ~P A | ( ( A \ x ) e. Fin \/ x = (/) ) } = { x e. ~P A | ( ( A \ x ) ~< _om \/ x = (/) ) } |
| 4 | fctop | |- ( A e. V -> { x e. ~P A | ( ( A \ x ) e. Fin \/ x = (/) ) } e. ( TopOn ` A ) ) |
|
| 5 | 3 4 | eqeltrrid | |- ( A e. V -> { x e. ~P A | ( ( A \ x ) ~< _om \/ x = (/) ) } e. ( TopOn ` A ) ) |