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[Merged by Bors] - chore(topology/compacts): inhabit instances #4462

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[Merged by Bors] - chore(topology/compacts): inhabit instances #4462

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4f1efbb
chore(topology/compacts): inhabit instances
hrmacbeth Oct 6, 2020
f111c0f
better inhabitation
hrmacbeth Oct 6, 2020
0b28618
remove #lint
hrmacbeth Oct 6, 2020
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8 changes: 8 additions & 0 deletions src/topology/compacts.lean
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Expand Up @@ -27,6 +27,9 @@ namespace topological_space
/-- The type of closed subsets of a topological space. -/
def closeds := {s : set α // is_closed s}

/-- The type of closed subsets is inhabited, with default element the empty set. -/
instance : inhabited (closeds α) := ⟨⟨∅, is_closed_empty ⟩⟩

/-- The compact sets of a topological space. See also `nonempty_compacts`. -/
def compacts : Type* := { s : set α // is_compact s }

Expand All @@ -35,6 +38,11 @@ non-emptiness will be useful in metric spaces, as we will be able to put
a distance (and not merely an edistance) on this space. -/
def nonempty_compacts := {s : set α // s.nonempty ∧ is_compact s}

/-- In an inhabited space, the type of nonempty compact subsets is also inhabited, with
default element the singleton set containing the default element. -/
instance nonempty_compacts_inhabited [inhabited α] : inhabited (nonempty_compacts α) :=
⟨⟨{default α}, singleton_nonempty (default α), compact_singleton ⟩⟩

/-- The compact sets with nonempty interior of a topological space. See also `compacts` and
`nonempty_compacts`. -/
@[nolint has_inhabited_instance]
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