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Metamath Proof Explorer
Description: The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013)
|
|
Ref |
Expression |
|
Hypothesis |
mopnval.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
|
Assertion |
mopntop |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mopnval.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 2 |
1
|
mopntopon |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 3 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
| 4 |
2 3
|
syl |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top ) |