View source for Monotonically normal space

==Definition==

===Definition with symbols===

A [[topological space]] <math>X</math> is termed '''monotonically normal''' if it is a [[T1 space]] (i.e., all points are closed) and there exists an operator <math>G</math> from ordered pairs of disjoint closed sets to open sets, such that:

# For any disjoint closed subsets <math>A,B</math>, <math>G(A,B)</math> contains <math>A</math> and its closure is disjoint from <math>B</math>
# If <math>A \subseteq A'</math> and <math>B' \subseteq B</math> with all four sets being closed, <math>A</math> disjoint from <math>B</math>, and <math>A'</math> disjoint from <math>B'</math>, we have:

<math>G(A,B) \subseteq G(A',B')</math>

This is the ''monotonicity'' condition. Such an operator <math>G</math> is termed a monotone normality operator.

{{topospace property}}

{{variationof|normality}}

==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::metrizable space]] || underlying topology of a [[metric space]] || [[metrizable implies monotonically normal]] || [[monotonically normal not implies metrizable]] || {{intermediate notions short|monotonically normal space|metrizable space}}
|-
| [[Weaker than::ordered field-metrizable space]] || underlying topology of a space with a metric taking values in an ordered field || [[ordered field-metrizable implies monotonically normal]] || [[monotonically normal not implies ordered field-metrizable]] || {{intermediate notions short|monotonically normal space|ordered field-metrizable space}}
|-
| [[Weaker than::linearly orderable space]] || [[order topology]] from a linear ordering on a set || [[linearly orderable implies monotonically normal]] || [[monotonically normal not implies linearly orderable]] || {{intermediate notions short|monotonically normal space|linearly orderable space}}
|-
| [[Weaker than::elastic space]] || || [[elastic implies monotonically normal]] || [[monotonically normal not implies elastic]] || {{intermediate notions short|monotonically normal space|elastic space}}
|-
| [[Weaker than::closed sub-Euclidean space]] || || (via metrizable) || (via metrizable) || {{intermediate notions short|monotonically normal space|closed sub-Euclidean space}}
|-
| [[Weaker than::manifold]] || || (via metrizable) || (via metrizable) || {{intermediate notions short|monotonically normal space|manifold}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::normal space]] || any two disjoint closed subsets are separated by disjoint open subsets || [[monotonically normal implies normal]] || [[normal not implies monotonically normal]] || {{intermediate notions short|normal space|monotonically normal space}}
|-
| [[Stronger than::hereditarily normal space]] || every subspace is a [[normal space]] || [[monotonically normal implies hereditarily normal]] || [[hereditarily normal implies monotonically normal]] || {{intermediate notions short|hereditarily normal space|monotonically normal space}}
|-
| [[Stronger than::collectionwise normal space]] || every discrete collection of closed subsets can be separated by disjoint open subsets || [[monotonically normal not implies collectionwise normal]] || [[collectionwise normal not implies monotonically normal]] || {{intermediate notions short|collectionwise normal space|monotonically normal space}}
|-
| [[Stronger than::hereditarily collectionwise normal space]] || every subspace is [[collectionwise normal space|collectionwise normal]] || [[monotonically normal implies hereditarily collectionwise normal]] || [[hereditarily collectionwise normal not implies monotonically normal]] || {{intermediate notions short|hereditarily collectionwise normal space|monotonically normal space}}
|-
| [[Stronger than::completely regular space]] || || (via normal) || (via normal) || {{intermediate notions short|completely regular space|monotonically normal space}}
|-
| [[Stronger than::regular space]] || || (via normal) || (via normal) || {{intermediate notions short|regular space|monotonically normal space}}
|-
| [[Stronger than::Hausdorff space]]|| || (via normal) || (via normal) || {{intermediate notions short|Hausdorff space|monotonically normal space}}
|-
| [[Stronger than::Urysohn space]] || || (via normal) || (via normal) || {{intermediate notions short|Urysohn space|monotonically normal space}}
|-
| [[Stronger than::collectionwise Hausdorff space]] || || (via collectionwise normal) || (via collectionwise normal) || {{intermediate notions short|collectionwise Hausdorff space|monotonically normal space}}
|}

===Incomparable properties===

* [[Perfectly normal space]]

==Metaproperties==

{{subspace-closed}}

Any subspace of a monotonically normal space is monotonically normal. {{proofat|[[Monotone normality is hereditary]]}}