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	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
Assertion	unimopn	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge A\subseteq J\right)\to \bigcup A\in J$$

Proof

Step	Hyp	Ref	Expression
1		mopni.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
2	1	mopntop	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to J\in \mathrm{Top}$$
3		uniopn	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq J\right)\to \bigcup A\in J$$
4	2 3	sylan	$$⊢\left(D\in \mathrm{\infty Met}\left(X\right)\wedge A\subseteq J\right)\to \bigcup A\in J$$