Theorem mopntopon

Description: The set of open sets of a metric space X is a topology on X . Remark in Kreyszig p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015)

	Ref	Expression
Hypothesis	mopnval.1	|- J = ( MetOpen ` D )
Assertion	mopntopon	|- ( D e. ( *Met ` X ) -> J e. ( TopOn ` X ) )

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	|- J = ( MetOpen ` D )
2	1	mopnval	|- ( D e. ( *Met ` X ) -> J = ( topGen ` ran ( ball ` D ) ) )
3		blbas	|- ( D e. ( *Met ` X ) -> ran ( ball ` D ) e. TopBases )
4		tgtopon	|- ( ran ( ball ` D ) e. TopBases -> ( topGen ` ran ( ball ` D ) ) e. ( TopOn ` U. ran ( ball ` D ) ) )
5	3 4	syl	|- ( D e. ( *Met ` X ) -> ( topGen ` ran ( ball ` D ) ) e. ( TopOn ` U. ran ( ball ` D ) ) )
6		unirnbl	|- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )
7	6	fveq2d	|- ( D e. ( *Met ` X ) -> ( TopOn ` U. ran ( ball ` D ) ) = ( TopOn ` X ) )
8	5 7	eleqtrd	|- ( D e. ( *Met ` X ) -> ( topGen ` ran ( ball ` D ) ) e. ( TopOn ` X ) )
9	2 8	eqeltrd	|- ( D e. ( *Met ` X ) -> J e. ( TopOn ` X ) )