Normal space: Difference between revisions

Latest revision as of 02:27, 20 April 2016

Please see Convention:Hausdorffness assumption

Definition

Equivalent definitions in tabular format

No.	Shorthand	A topological space is said to be normal(-minus-Hausdorff) if ...	A topological space $X$ is said to be normal(-minus-Hausdorff) if ...
1	separation of disjoint closed subsets by open subsets	given any two disjoint closed subsets in the topological space, there are disjoint open sets containing them.	given any two closed subsets $A, B \subseteq X$ such that $A \cap B = \emptyset$ , there exist disjoint open subsets $U, V$ of $X$ such that $A \subseteq U, B \subseteq V$ , and $U \cap V = \emptyset$ .
2	separation of disjoint closed subsets by continuous functions	given any two disjoint closed subsets, there is a continuous function taking the value $0$ at one closed set and 1 at the other.	for any two closed subsets $A, B \subseteq X$ , such that $A \cap B = \emptyset$ , there exists a continuous map $f : X \to [0, 1]$ (to the closed unit interval) such that $f (x) = 0 \forall x \in A$ and $f (x) = 1 \forall x \in B$ .
3	point-finite open cover has shrinking	every point-finite open cover possesses a shrinking.	for any point-finite open cover $U_{i}, i \in I$ of $X$ , there exists a shrinking $V_{i}, i \in I$ : the $V_{i}$ form an open cover and $\bar{V_{i}} \subseteq U_{i}$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal Hausdorff space	a normal space that is also Hausdorff. Some people include Hausdorffness as part of the definition of normal space.	(obvious)	a set of size more than one with the trivial topology is normal but not Hausdorff	\|FULL LIST, MORE INFO
compact Hausdorff space	a compact space that is also Hausdorff	compact Hausdorff implies normal	normal not implies compact	Binormal space, Paracompact Hausdorff space\|FULL LIST, MORE INFO
hereditarily normal space	every subspace is a normal space under the subspace topology		normality is not hereditary	\|FULL LIST, MORE INFO
paracompact Hausdorff space	a paracompact space that is also Hausdorff	paracompact Hausdorff implies normal	normal not implies paracompact	Binormal space\|FULL LIST, MORE INFO
regular Lindelof space	both a regular space and a Lindelof space	regular Lindelof implies normal	normal not implies Lindelof	\|FULL LIST, MORE INFO
perfectly normal space	every closed subset is a G-delta subset	perfectly normal implies normal	normal not implies perfectly normal	Hereditarily normal space\|FULL LIST, MORE INFO
metrizable space	can be given the structure of a metric space with the same topology	metrizable implies normal	normal not implies metrizable	Collectionwise normal space, Elastic space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space, Paracompact Hausdorff space, Perfectly normal space, Protometrizable space\|FULL LIST, MORE INFO
CW-space	the underlying topological space of a CW-complex	CW implies normal	normal not implies CW	Hereditarily normal space, Paracompact Hausdorff space, Perfectly normal space\|FULL LIST, MORE INFO
linearly orderable space	obtained using the order topology for some linear ordering	linearly orderable implies normal	normal not implies linearly orderable	Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space\|FULL LIST, MORE INFO
collectionwise normal space
monotonically normal space
ultraconnected space		ultraconnected implies normal		\|FULL LIST, MORE INFO

@@ Line 4: / Line 4: @@
 ===Equivalent definitions in tabular format===
-Note that under some conventions, the definition below is taken as the definition of ''normal space''. However, under the convention followed in this wiki, the definition of [[normal space]] includes the assumption that points are closed, forcing the space to be a [[T1 space]] (and further also a [[Hausdorff space]], based on the definition). The definition given here is obtained when we relax the requirement of points being closed. Note that this definition ''includes'' the normal spaces where all points are closed, but also includes some other spaces.
 {| class="sortable" border="1"
@@ Line 15: / Line 13: @@
 |-
 | 3 || point-finite open cover has shrinking || every [[defining ingredient::point-finite collection|point-finite]] [[open cover]] possesses a [[shrinking]]. || for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.
+|}
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::normal Hausdorff space]] || a normal space that is also [[Hausdorff space|Hausdorff]]. Some people include Hausdorffness as part of the definition of normal space. || (obvious) || a set of size more than one with the trivial topology is normal but not Hausdorff || {{intermediate notions short|normal space|normal Hausdorff space}}
+|-
+| [[Weaker than::compact Hausdorff space]] || a [[compact space]] that is also [[Hausdorff space|Hausdorff]] || [[compact Hausdorff implies normal]] || [[normal not implies compact]] || {{intermediate notions short|normal space|compact Hausdorff space}}
+|-
+| [[Weaker than::hereditarily normal space]] || every subspace is a [[normal space]] under the [[subspace topology]] || || [[normality is not hereditary]] || {{intermediate notions short|normal space|hereditarily normal space}}
+|-
+| [[Weaker than::paracompact Hausdorff space]] || a [[paracompact space]] that is also [[Hausdorff space|Hausdorff]] || [[paracompact Hausdorff implies normal]] || [[normal not implies paracompact]] || {{intermediate notions short|normal space|paracompact Hausdorff space}}
+|-
+| [[Weaker than::regular Lindelof space]] || both a [[regular space]] and a [[Lindelof space]] || [[regular Lindelof implies normal]] || [[normal not implies Lindelof]] || {{intermediate notions short|normal space|regular Lindelof space}}
+|-
+| [[Weaker than::perfectly normal space]] || every [[closed subset]] is a [[G-delta subset]] || [[perfectly normal implies normal]] || [[normal not implies perfectly normal]] || {{intermediate notions short|normal space|perfectly normal space}}
+|-
+| [[Weaker than::metrizable space]] || can be given the structure of a [[metric space]] with the same topology || [[metrizable implies normal]] || [[normal not implies metrizable]] || {{intermediate notions short|normal space|metrizable space}}
+|-
+| [[Weaker than::CW-space]] || the underlying topological space of a [[CW-complex]] || [[CW implies normal]] || [[normal not implies CW]] || {{intermediate notions short|normal space|CW-space}}
+|-
+| [[Weaker than::linearly orderable space]] || obtained using the order topology for some linear ordering || [[linearly orderable implies normal]] || [[normal not implies linearly orderable]] || {{intermediate notions short|normal space|linearly orderable space}}
+|-
+| [[Weaker than::collectionwise normal space]] || || || ||
+|-
+| [[Weaker than::monotonically normal space]] || || || ||
+|-
+| [[Weaker than::ultraconnected space]]|| || [[ultraconnected implies normal]] || || {{intermediate notions short|normal space|ultraconnected space}}
 |}