Description: The union of a collection of open sets of a metric space is open. Theorem T2 of Kreyszig p. 19. (Contributed by NM, 4-Sep-2006) (Revised by Mario Carneiro, 23-Dec-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | mopni.1 | |- J = ( MetOpen ` D ) |
|
| Assertion | unimopn | |- ( ( D e. ( *Met ` X ) /\ A C_ J ) -> U. A e. J ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mopni.1 | |- J = ( MetOpen ` D ) |
|
| 2 | 1 | mopntop | |- ( D e. ( *Met ` X ) -> J e. Top ) |
| 3 | uniopn | |- ( ( J e. Top /\ A C_ J ) -> U. A e. J ) |
|
| 4 | 2 3 | sylan | |- ( ( D e. ( *Met ` X ) /\ A C_ J ) -> U. A e. J ) |