Theorem mopn0

Description: The empty set is an open set of a metric space. Part of Theorem T1 of Kreyszig p. 19. (Contributed by NM, 4-Sep-2006)

	Ref	Expression
Hypothesis	mopni.1	$$⊢J=\mathrm{MetOpen}\left(D\right)$$
Assertion	mopn0	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to \varnothing \in J$$

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		0opn	$$⊢J\in \mathrm{Top}\to \varnothing \in J$$
4	2 3	syl	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to \varnothing \in J$$