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Metamath Proof Explorer
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 = ( MetOpen ‘ 𝐷 ) |
|
Assertion |
mopn0 |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∅ ∈ 𝐽 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mopni.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 2 |
1
|
mopntop |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top ) |
| 3 |
|
0opn |
⊢ ( 𝐽 ∈ Top → ∅ ∈ 𝐽 ) |
| 4 |
2 3
|
syl |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∅ ∈ 𝐽 ) |