Separation axiom metaproperty chart: Difference between revisions

Latest revision as of 20:37, 21 July 2008

Property	T family name	Hereditary	Weakly hereditary	Product-closed	Refining-preserved	Local
Kolmogorov	$T_{0}$	Yes	Yes	Yes	Yes	Yes
T1	$T_{1}$	Yes	Yes	Yes	Yes	Yes
Hausdorff	$T_{2}$	Yes	Yes	Yes	Yes	No
Urysohn	$T_{2.5}$	Yes	Yes	Yes	Yes	No
Regular	$T_{3}$	Yes	Yes	Yes	No	No
Completely regular	$T_{3.5}$	Yes	Yes	Yes	No	No
Normal	$T_{4}$	No	Yes	No	No	No
Hereditarily normal	$T_{5}$	Yes	Yes	No	No	No
Perfectly normal	$T_{6}$	Yes	Yes	?	?	No
Monotonically normal	-	Yes	Yes	?	?	No
Collectionwise Hausdorff	-	Yes	Yes	?	Yes	No

@@ Line 4: / Line 4: @@
 |[[Kolmogorov space|Kolmogorov]] || <math>T_0</math> || Yes || Yes || Yes || Yes || Yes
 |-
-| [[T1 space|T1]] || <math>T_1</math> || Yes || Yes || Yes || Yes || Yes
+| [[T1 space|T1]] || <math>T_1</math> || [[T1 is hereditary|Yes]] || Yes || [[T1 is product-closed|Yes]] || [[T1 is refining-preserved|Yes]] || Yes
 |-
-| [[Hausdorff space|Hausdorff]] || <math>T_2</math> || [[Hausdorffness is hereditary|Yes]] || Yes || [[Hausdorffness is product-closed|Yes]] || Yes || No
+| [[Hausdorff space|Hausdorff]] || <math>T_2</math> || [[Hausdorffness is hereditary|Yes]] || Yes || [[Hausdorffness is product-closed|Yes]] || [[Hausdorffness is refining-preserved|Yes]] || No
 |-
-| [[Urysohn space|Urysohn]] || <math>T_{2.5}</math> || Yes || Yes || Yes || Yes || No
+| [[Urysohn space|Urysohn]] || <math>T_{2.5}</math> || [[Urysohn property is hereditary|Yes]] || Yes || Yes || Yes || No
 |-
-| [[Regular space|Regular]] || <math>T_3</math> || Yes || Yes || Yes || No || No
+| [[Regular space|Regular]] || <math>T_3</math> || [[Regularity is hereditary|Yes]] || Yes || [[Regularity is product-closed|Yes]] || [[Regularity is not refining-preserved|No]] || No
 |-
-| [[Completely regular space|Completely regular]] || <math>T_{3.5}</math> || Yes || Yes || Yes || No || No
+| [[Completely regular space|Completely regular]] || <math>T_{3.5}</math> || [[Complete regularity is hereditary|Yes]] || Yes || [[Complete regularity is product-closed|Yes]] || [[Complete regularity is not refining-preserved|No]] || No
 |-
 | [[Normal space|Normal]] || <math>T_4</math> || No || [[Normality is weakly hereditary|Yes]] || No || No || No
@@ Line 22: / Line 22: @@
 | [[Monotonically normal space|Monotonically normal]] || - || Yes || Yes || ? || ? || No
 |-
-| [[Collectionwise Hausdorff space|Collectionwise Hausdorff]] || Yes || Yes || ? || Yes || No
+| [[Collectionwise Hausdorff space|Collectionwise Hausdorff]] || - || Yes || Yes || ? || Yes || No
 |-
 |}