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 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	mopnval	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 = ( topGen ‘ ran ( ball ‘ 𝐷 ) ) )
3		blbas	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ∈ TopBases )
4		tgtopon	⊢ ( ran ( ball ‘ 𝐷 ) ∈ TopBases → ( topGen ‘ ran ( ball ‘ 𝐷 ) ) ∈ ( TopOn ‘ ∪ ran ( ball ‘ 𝐷 ) ) )
5	3 4	syl	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( topGen ‘ ran ( ball ‘ 𝐷 ) ) ∈ ( TopOn ‘ ∪ ran ( ball ‘ 𝐷 ) ) )
6		unirnbl	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∪ ran ( ball ‘ 𝐷 ) = 𝑋 )
7	6	fveq2d	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( TopOn ‘ ∪ ran ( ball ‘ 𝐷 ) ) = ( TopOn ‘ 𝑋 ) )
8	5 7	eleqtrd	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( topGen ‘ ran ( ball ‘ 𝐷 ) ) ∈ ( TopOn ‘ 𝑋 ) )
9	2 8	eqeltrd	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )