Theorem unimopn

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 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝐽 ) → ∪ 𝐴 ∈ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	mopntop	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top )
3		uniopn	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ) → ∪ 𝐴 ∈ 𝐽 )
4	2 3	sylan	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝐽 ) → ∪ 𝐴 ∈ 𝐽 )