Description: An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | xmstps | ⊢ ( 𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp ) |
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 ) |