Theorem xmstps

Description: An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015)

	Ref	Expression
Assertion	xmstps	⊢ ( 𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( TopOpen ‘ 𝑀 ) = ( TopOpen ‘ 𝑀 )
2		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
3		eqid	⊢ ( ( dist ‘ 𝑀 ) ↾ ( ( Base ‘ 𝑀 ) × ( Base ‘ 𝑀 ) ) ) = ( ( dist ‘ 𝑀 ) ↾ ( ( Base ‘ 𝑀 ) × ( Base ‘ 𝑀 ) ) )
4	1 2 3	isxms	⊢ ( 𝑀 ∈ ∞MetSp ↔ ( 𝑀 ∈ TopSp ∧ ( TopOpen ‘ 𝑀 ) = ( MetOpen ‘ ( ( dist ‘ 𝑀 ) ↾ ( ( Base ‘ 𝑀 ) × ( Base ‘ 𝑀 ) ) ) ) ) )
5	4	simplbi	⊢ ( 𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp )