Theorem mopntop

Description: The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypothesis	mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion	mopntop	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	mopntopon	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
3		topontop	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top )
4	2 3	syl	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top )