File: Mathlib/Topology/Defs/Sequences.lean

full_name: seqClosure
code: /-- The sequential closure of a set `s : Set X` in a topological space `X` is the set of all `a : X`
which arise as limit of sequences in `s`. Note that the sequential closure of a set is not
guaranteed to be sequentially closed. -/
def seqClosure (s : Set X) : Set X 
goal: Set X

full_name: IsSeqClosed
code: /-- A set `s` is sequentially closed if for any converging sequence `x n` of elements of `s`, the
limit belongs to `s` as well. Note that the sequential closure of a set is not guaranteed to be
sequentially closed. -/
def IsSeqClosed (s : Set X) : Prop 
goal: Prop

full_name: SeqContinuous
code: /-- A function between topological spaces is sequentially continuous if it commutes with limit of
 convergent sequences. -/
def SeqContinuous (f : X → Y) : Prop 
goal: Prop

full_name: IsSeqCompact
code: /-- A set `s` is sequentially compact if every sequence taking values in `s` has a
converging subsequence. -/
def IsSeqCompact (s : Set X) 
goal: Prop

full_name: SeqCompactSpace
code: /-- A space `X` is sequentially compact if every sequence in `X` has a
converging subsequence. -/
@[mk_iff]
class SeqCompactSpace : Prop where
  isSeqCompact_univ : IsSeqCompact (univ : Set X)
goal: Prop

full_name: SeqCompactSpace.isSeqCompact_univ
code: /-- A space `X` is sequentially compact if every sequence in `X` has a
converging subsequence. -/
@[mk_iff]
class SeqCompactSpace : Prop where
  isSeqCompact_univ : IsSeqCompact (univ : Set X)
goal: IsSeqCompact Set.univ

full_name: seqCompactSpace_iff
code: /-- A space `X` is sequentially compact if every sequence in `X` has a
converging subsequence. -/
@[mk_iff]
class SeqCompactSpace : Prop where
  isSeqCompact_univ : IsSeqCompact (univ : Set X)
goal: SeqCompactSpace X ↔ IsSeqCompact Set.univ

full_name: FrechetUrysohnSpace
code: /-- A topological space is called a *Fréchet-Urysohn space*, if the sequential closure of any set
is equal to its closure. Since one of the inclusions is trivial, we require only the non-trivial one
in the definition. -/
class FrechetUrysohnSpace : Prop where
  closure_subset_seqClosure : ∀ s : Set X, closure s ⊆ seqClosure s
goal: Prop

full_name: FrechetUrysohnSpace.closure_subset_seqClosure
code: /-- A topological space is called a *Fréchet-Urysohn space*, if the sequential closure of any set
is equal to its closure. Since one of the inclusions is trivial, we require only the non-trivial one
in the definition. -/
class FrechetUrysohnSpace : Prop where
  closure_subset_seqClosure : ∀ s : Set X, closure s ⊆ seqClosure s
goal: closure s ⊆ seqClosure s

full_name: SequentialSpace.isClosed_of_seq
code: /-- A topological space is said to be a *sequential space* if any sequentially closed set in this
space is closed. This condition is weaker than being a Fréchet-Urysohn space. -/
class SequentialSpace : Prop where
  isClosed_of_seq : ∀ s : Set X, IsSeqClosed s → IsClosed s
goal: IsSeqClosed s → IsClosed s

full_name: SequentialSpace
code: /-- A topological space is said to be a *sequential space* if any sequentially closed set in this
space is closed. This condition is weaker than being a Fréchet-Urysohn space. -/
class SequentialSpace : Prop where
  isClosed_of_seq : ∀ s : Set X, IsSeqClosed s → IsClosed s
goal: Prop

full_name: IsSeqClosed.isClosed
code: /-- In a sequential space, a sequentially closed set is closed. -/
protected theorem IsSeqClosed.isClosed [SequentialSpace X] {s : Set X} (hs : IsSeqClosed s) :
    IsClosed s 
goal: IsClosed s