Metamath Proof Explorer


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 )