title: Monotonically normal space
text: # Monotonically normal space
  
  In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesting properties; for example metric spaces and linearly ordered spaces are monotonically normal, and every monotonically normal space is hereditarily normal.
  
  ## Definition
  
  A topological space $X$ is called monotonically normal if it satisfies any of the following equivalent definitions:
  
  ### Definition 1
  
  The space $X$ is $T_1$ and there is a function $G$ that assigns to each ordered pair $(A,B)$ of disjoint closed sets in $X$ an open set $G(A,B)$ such that:
  
      (i) $A\subseteq G(A,B)\subseteq {\overline {G(A,B)}}\subseteq X\setminus B$;
      (ii) $G(A,B)\subseteq G(A',B')$ whenever $A\subseteq A'$ and $B'\subseteq B$.
  
  Condition (i) says $X$ is a normal space, as witnessed by the function $G$.
  
  Condition (ii) says that $G(A,B)$ varies in a monotone fashion, hence the terminology monotonically normal. The operator $G$ is called a monotone normality operator.
  
  One can always choose $G$ to satisfy the property
  
      $G(A,B)\cap G(B,A)=\emptyset$,
  
  by replacing each $G(A,B)$ by $G(A,B)\setminus {\overline {G(B,A)}}$.
  
  ### Definition 2
  
  The space $X$ is $T_1$ and there is a function $G$ that assigns to each ordered pair $(A,B)$ of separated sets in $X$ (that is, such that $A\cap {\overline {B}}=B\cap {\overline {A}}=\emptyset$) an open set $G(A,B)$ satisfying the same conditions (i) and (ii) of Definition 1.
  
  ### Definition 3
  
  The space $X$ is $T_1$ and there is a function $\mu$ that assigns to each pair $(x,U)$ with $U$ open in $X$ and $x\in U$ an open set $\mu (x,U)$ such that:
  
      (i) $x\in \mu (x,U)$;
      (ii) if $\mu (x,U)\cap \mu (y,V)\neq \emptyset$, then $x\in V$ or $y\in U$.
  
  Such a function $\mu$ automatically satisfies
  
      $x\in \mu (x,U)\subseteq {\overline {\mu (x,U)}}\subseteq U$.
  
  (Reason: Suppose $y\in X\setminus U$. Since $X$ is $T_1,$ there is an open neighborhood $V$ of $y$ such that $x\notin V$. By condition (ii), $\mu (x,U)\cap \mu (y,V)=\emptyset$, that is, $\mu (y,V)$ is a neighborhood of $y$ disjoint from $\mu (x,U)$. So $y\notin {\overline {\mu (x,U)}}$.)
  
  ### Definition 4
  
  Let ${\mathcal {B}}$ be a base for the topology of $X$. The space $X$ is $T_1$ and there is a function $\mu$ that assigns to each pair $(x,U)$ with $U\in {\mathcal {B}}$ and $x\in U$ an open set $\mu (x,U)$ satisfying the same conditions (i) and (ii) of Definition 3.
  
  ### Definition 5
  
  The space $X$ is $T_1$ and there is a function $\mu$ that assigns to each pair $(x,U)$ with $U$ open in $X$ and $x\in U$ an open set $\mu (x,U)$ such that:
  
      (i) $x\in \mu (x,U)$;
      (ii) if $U$ and $V$ are open and $x\in U\subseteq V$, then $\mu (x,U)\subseteq \mu (x,V)$;
      (iii) if $x$ and $y$ are distinct points, then $\mu (x,X\setminus \{y\})\cap \mu (y,X\setminus \{x\})=\emptyset$.
  
  Such a function $\mu$ automatically satisfies all conditions of Definition 3.
  
  ## Examples
  
  * Every metrizable space is monotonically normal.
  * Every linearly ordered topological space (LOTS) is monotonically normal. This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.
  * The Sorgenfrey line is monotonically normal. This follows from Definition 4 by taking as a base for the topology all intervals of the form $[a,b)$ and for $x\in [a,b)$ by letting $\mu (x,[a,b))=[x,b)$. Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the double arrow space.
  * Any generalised metric is monotonically normal.
  
  ## Properties
  
  * Monotone normality is a hereditary property: Every subspace of a monotonically normal space is monotonically normal.
  * Every monotonically normal space is completely normal Hausdorff (or $T_5).$
  * Every monotonically normal space is hereditarily collectionwise normal.
  * The image of a monotonically normal space under a continuous closed map is monotonically normal.
  * A compact Hausdorff space $X$ is the continuous image of a compact linearly ordered space if and only if $X$ is monotonically normal.
  