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In this note we give definitions from Morillon [1988] for forms [14 L], [14 BP] through [14 BX],[14 CC] through [14 CH], [118 I] through [118 T], Form 331, Form 332, Form 343, Form 344 ,and [345 C] through [345 E].

Content:

In this note we give definitions from Morillon [1988] for forms [14 L], [14 BP] through [14 BX],[14 CC] through [14 CH], [118 I] through [118 T], Form 331, Form 332, Form 343, Form 344 ,and [345 C] through [345 E].

Let \(X\) be a topological space, let \(E = \{(A,B) : A \) and \(B\) are closed in \(X\) and \(A\cap B = \emptyset\}\) and let \(F =\{(U,V) : U \) and \(V\) are open in \(X\) and \(U \cap V = \emptyset\}\).

1. \(f:E\to F\) is a normality operator on \(X\) if \(\forall(A,B)\in E\), \(f(A,B) = (U,V)\) where \(A\subseteq U\) and \(B\subseteq V\).
2. \(X\) is effectively normal if \(X\) is \(T_2\) and there is a normality operator on \(X\).
3. \(X\) is hereditarily normal (or completely normal) if all subspaces of \(X\) are normal.
4. \(X\) is hereditarily effectively normal if all subspaces of \(X\) are effectively normal.
5. A set \(I \subseteq \cal P(X)\) is discrete if \(\forall x\in X\), \(\exists\) a neighborhood \(N\) of \(x\) such that \(N\cap A\ne\emptyset\) for at most one element \(A\in I\).
6. \(X\) is collectionwise normal if \(X\) is \(T_2\) and for every discrete \(I\subseteq \{U: U\) is closed in \(S\}\) there is a set \(J\) of pairwise disjoint open subsets of \(X\) such that for every\(F\in I\), \(\exists U\in J\) such that \(F\subseteq U\).
7. Let \(D = \{I: I\) is a discrete collection of closed subsets of \(X\}\) and \(O = \{J: J\) is a collection of pairwise disjoint open subsets of \(X\}\).  Then \(f: D\to O\) is a collectionwise normality operator on \(X\) if \(\forall I\in D\), if\(f(I) = J\) then \(\forall F\in I\), \(\exists U\in J\) such that \(F\subseteq U\).  \(N_c(X)\) denotes the set of collectionwise normality operators on \(X\).
8. \(X\) is effectively collectionwise normal if X is \(T_2\)and \(N_c(X) \ne \emptyset\).
9. \(X\) is highly separated if \(X\) is \(T_2\) and \(\{N_c(F):F\subseteq X\}\) has a choice function.
10. \(X\) is functionally separated if for all pairs \((A,B)\) of closed disjoint subsets of \(X\) there is a function \(f: X\to [0,1]\) such that \(A\subseteq f^{-1}(0)\) and \(B\subseteq f^{-1}(1)\).
11. \(X\) is Tietze-Urysohn if for every closed \(F\subseteq X\) and every continuous function \(f: F\to [0,1]\) there is a continuous function \(g: X \to [0,1]\) which extends \(f\).
12. Let \(C = \{g : g:X\to [0,1]\) and \(g\) is continuous\(\}\). A function \(f: E\to C\) is a Urysohn operator on \(X\) if\(\forall (A,B)\in E\), if \(U = f(A,B)\) then \(A \subseteq U^{-1}(0)\) and\(B\subseteq U^{-1}(1)\). \(U(X)\) denotes the set of Urysohn operators on\(X\).
13. Let \(B = \{(F,U) : F\) is a closed subset of \(X\) and \(U:F\to [0,1]\) is continuous\(\}\). A function \(f: B\to C\) is a Tietze-Urysohn operator if \(\forall (F,U)\in B\), if \(f(F,U) = V\) then \(V\) extends \(U\).
14. \(X\) is effectively functionally separated if \(U(X) \ne\emptyset\). \(X\) is effectively Tietze-Urysohn if T-U\((X) \ne\emptyset\).  (Where T-U\((X)\) is the set of all Tietze-Urysohn operatorson \(X\).)
15. \(X\) is highly functionally separated if \(\{U(A) : A\subseteq X\}\) has a choice function.
16. \(X\) is highly Tietze-Urysohn if \(\{\)T-U\((A): A\subseteq X\}\) has a choice function.
17. \(X\) is collectionwise separated (or collectionwise Hausdorff) if it is \(T_2\) and for every discrete set \(I\) of singletons from \(X\) there is a set \(J\) of open subsets of \(X\) such that for every\(F\in I\), there is a unique \(O\in J\) containing (the singleton) \(F\).
    Assume that \((X,\le)\) is a linear order.
18. \(A\subseteq X\) is convex if \(\forall (x,y)\in A^2\),\(\forall u\in X\), if \(x < u < y\) then \(u\in A\).
19. \((X,\le)\) is conditionally complete if each non-empty subset with an upper bound has a least upper bound and each non-empty subset with a lower bound has a greatest lower bound.
20. \((X,\le)\) is complete if \(X=\emptyset\) or every subset has a least upper bound and a greatest lower bound.
21. Let \(X\) be a topological space, let \(E\) be defined as above and let \(\Omega(X)\) be the set of open subsets of \(X\).  A function\(f: E\to \Omega(X)\) is a monotone normality operator on \(X\) if for every \((F,G)\in E\) the following conditions hold
    1. \(F\subseteq f(F,G) \subseteq \overline{f(F,G)} \subseteq X-G\).
    2. If \((F',G')\in E\), \(F\subseteq F'\) and \(G'\subseteq G\)then \(f(F,G)\subseteq f(F',G')\).
22. A topological space \(X\) is monotonely normal if itis \(T_1\) and there is a monotone normality operator on \(X\).
23. Let \((X,\le)\) be a complete linear order and let \(E\) be the set of non-empty open intervals of \(X\).  An increasing choice function on \((X,\le)\) is a function \(*: E\to X\) such that
    1. \(\forall I\in E\), \(*(I)\in I\).
    2. If \((a,b)\in E\) and \((b,c)\in E\) then \(*((a,b))\le*((a,c)) \le *((b,c))\).
24. A topological space \(X\) is completely regular if for all open \(U\subseteq X\) and all \(p\in U\), there is a continuous \(f: X\to [0,1]\) such that \(f(p) = 0\) and \(\forall x\in X - U\), \(f(x) = 1\).
25. An ordered set \((T,\le)\) is a lattice if any two elements have a greatest lower bound and a least upper bound.  A lattice is bounded if it has a smallest and largest element.
26. An ideal \(I\) in a lattice \((T,\le)\) is prime if \(\forall x,y \in T\), if \(x\land y \in I\) then either \(x\in I\) or \(y\in I\).
27. If \(T\) is a bounded distributive lattice a function \(f\) is a prime ideal choice function on \(T\) if for each proper ideal\(I\) of \(T\), \(f(I)\) is a prime ideal containing \(I\).
28. Assume that \(T\) is a bounded distributive lattice in which every subset has a least upper bound then \(T\) is \(\lor\)-compact if for all \(F\subseteq T\) such that \(\bigvee F = 1\), there is a finite\(F'\subseteq F\) such that \(\bigvee F' = 1\). (If \((\Omega,\subseteq)\) is the lattice of open sets of a topological space \(X\) then \(X\) is compact if and only if \((\Omega,\subseteq)\) is\(\lor\)-compact.)
29. Let \(E\) be a set.  An ultra filter choice function \(f\)on \(E\) is a function such that for all proper filters \(I\) in \(\cal P(E)\),\(f(I)\) is an ultra filter containing \(I\).
30. Let \(E\) be a set and \(U\) an ultra filter on a set \(I\). \(U\)is \(E\)-adequate if for every filter \(F\) in \(\cal P(E)\) there is a function \(g: I\to E\) such that \(g(U)\) contains \(F\).
31. A topological space is irreducible if it is not empty and is not the union of two proper closed subsets.
32. A topological space \(X\) is sober if every closed irreducible subset of \(X\) is the closure of a unique singleton.
33. A Banach algebra is a normed algebra in which every Cauchy filter converges.
34. Let \((I,\le)\) be a preorder (reflexive and transitive), let\((E_\alpha)_{\alpha\in I}\) be a family of sets and for each \((\alpha,\beta)\in I\times I\) let \(f_{\alpha,\beta}\) be a function from \(E_\beta\)to \(E_\alpha\).  Assume that the \(f_{\alpha,\beta}\) satisfy the following conditions:
    • \((LP_{I})\) \(\alpha\le\beta\le\gamma \Rightarrow f_{\alpha,\gamma} = f_{\alpha,\beta}\circ f_{\beta,\gamma}\).
    • \((LP_{II})\) For all \(\alpha\in I\), \(f_{\alpha,\alpha}\) is the identity on \(E_\alpha\).
    Then \(((E_\alpha)_{\alpha \in I},(f_{\alpha,\beta})_{(\alpha,\beta)\in I \times I})\) is called a projective system relative to the preorder\((I,\le)\).  Let \(E = \prod_{\alpha\in I} E_{\alpha}\) and for each pair\((\alpha,\beta)\in I\times I\) such that \(\alpha \le \beta\), let \(E_{\alpha,\beta} = \{x\in E: f_{\alpha,\beta}(x_\beta) =x_\alpha\}\). Then the set \(Y = \bigcap_{\alpha < \beta}E_{\alpha,\beta}\) is called the projective limit of the projective system \(((E_\alpha)_{\alpha\in I},(f_{\alpha,\beta})_{(\alpha,\beta)\in I \times I})\).  \(Y\) is denoted \(\lim_\leftarrow (E_\alpha,f_{\alpha,\beta})\).  In case the preorder \((I,\le)\) is directed to the right(that is, \(\forall (\alpha,\beta)\in I^2\), \(\exists \gamma\in I \) such that \(\alpha\le \gamma\) and \(\beta \le \gamma\).) we say that the projective system \(((E_\alpha)_{\alpha\in I},(f_{\alpha,\beta})_{(\alpha,\beta)\in I \times I})\) is directed.
35. If \(((X_\alpha)_{\alpha\in I},(f_{\alpha,\beta})_{(\alpha,\beta)\in I \times I})\) is a projective system relative to the preorder \((I,\le)\) and if each \(X_\alpha\) has a topology \(C_\alpha\) such that \(f_{\alpha,\beta}\) is continuous relative to the topologies \(C_\alpha\) and \(C_\beta\) then the system\(((X_\alpha)_{\alpha\in I},(f_{\alpha,\beta})_{(\alpha,\beta)\in I \times I})\) is called a projective system of topological spaces. In this case the projective limit of the system with the topology induced by the product topology on \(\prod_{\alpha\in I} X_\alpha\) is called the the projective limit of the projective system of topological spaces.
36. If \((A,\land,\lor)\) is a bounded, distributive lattice, \(X\)is a subset of \(A\) and \(F\) is a filter in \(A\), then \(F\) respects \(X\) if \(X\subseteq F\hbox{ and }\bigwedge X\) exists implies\(\bigwedge X \in F\).
37. If \((A,\land,\lor)\) is a bounded, distributive lattice, \(X\)is a subset of \(A\) and \(F\) is a filter in \(A\), then \(F\) strongly respects \(X\) if \(X\subseteq F\) implies \(\exists y\in F\) such that\(\forall x\in X\), \(y\le x\).
38. If \(X\) is a subset of a bounded, distributive lattice\((A,\land,\lor)\) then \(X\) is of Boolean type if for all\(b\in A\), \([(\forall x\in X, b\lor x = 1)\hbox{ and } (\forall y\le X, y\le b)]\) implies \(b=1\).  (\(y\le X\) means \((\forall x\in X)(y\le x)\).)
39. If \((A,\land,\lor)\) is a bounded, distributive lattice we say that \(A\) has the Rasiowa-Sikorski property  (respectively the strong Rasiowa-Sikorski property) if for all \(a,b\in A\) such that \(a\not\le b\) and all sequences \((X_n)_{n\in\omega}\) of subsets of \(A\) of Boolean type there is a prime filter containing \(a\),not containing \(b\) and respecting (respectively strongly respecting) each \(X_n\).
40. If \((A,\land,\lor)\) is a bounded distributive lattice then an element \(a\in A\) is \(\lor\)-compact if for every subset \(I\subseteq A\), \(\bigvee I = a\) implies \((\exists J\subseteq I)(J\hbox{ finite and }\bigvee J = a)\).
41. If \(X\) is a topological space, the Stone-Čech Compactification of \(X\), is the pair \((e,\beta(X))\), defined as follows: Let \(F(X)\) be the set of all continuous functions on \(X\) to the closed unit interval \(I\) and let \(e\) be a map from \(X\) to \(I^{F(X)}\) such that \(f\)-th coordinate of \(e(x)\) is \(f(x)\) for each \(f\in F(X)\).  Then \(\beta(X)\) is the closure of \(e[X]\) in the cube \(I^{F(X)}\).

Howard-Rubin number: 71

Type: Definitions

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