Separation axioms
Logic
Propositional logic
Predicate-Logic
Set
Set
Set operation
Binary relation and Map
Equivalence relation
Ordered set
Wellordered set
Axiom of Choice
Cardinality
Topology
Topology
Topological space
Continuous map
Compact
Separation axioms
Metric-space
Convergence
Linear algebra
Linear space
Product space
Basis
Orthogonal system
Linear operator
Determinant
Elementary matrix
Rank
Eigenspace
Nomal matrix
Group
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Quotient group
Homomorphism
Cartesian product
Group action
Real analysis
Convergence of real sequence
Convergence theorem1
Convergence of real series
Convergence of real function
Convergence theorem2
Measure theory
Finitely additive measure
Outer measure
Measure space
Back
en
ja
Separation axiom
(
T
1
)
∀
x
,
y
∈
X
(
x
=
y
)
,
∃
O
∈
O
(
x
∈
O
,
y
∈
/
O
)
(
T
2
)
∀
x
,
y
∈
X
(
x
=
y
)
,
∃
O
1
,
O
2
∈
O
(
x
∈
O
1
,
y
∈
O
2
,
O
1
∩
O
2
=
ϕ
)
(
T
3
)
∀
F
∈
C
,
∀
x
∈
/
F
,
∃
O
1
,
O
2
∈
O
(
x
∈
O
1
,
F
⊂
O
2
,
O
1
∩
O
2
=
ϕ
)
(
T
4
)
∀
F
1
,
F
2
∈
C
(
F
1
∩
F
2
=
ϕ
)
,
∃
O
1
,
O
2
∈
O
(
F
1
⊂
O
1
,
F
2
⊂
O
2
,
O
1
∩
O
2
=
ϕ
)
Hausdorff space
(
X
,
O
)
is a Hausdorff space:
(
T
2
)
Regular space
(
X
,
O
)
is a regular space:
(
T
1
)
,
(
T
3
)
Normal space
(
X
,
O
)
is a normal space:
(
T
1
)
,
(
T
4
)
Corollary
Normal
⇒
Regular
⇒
Hausdorff
⇒
(
T
1
)
Separation axiom
Hausdorff space
Regular space
Normal space
Corollary
