Statement:
Every linearly orderable topological space is normal. Birkhoff [1967], p 241.
Howard_Rubin_Number: 118
Parameter(s): This form does not depend on parameters
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Book references
Lattice Theory (3rd edition), Birkhoff, G., 1967
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 118 A | For each complete linear order, the set of non-emptyconvex subsets has a choice function. van Douwen [1985]. |
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| 118 B | For each complete linear order, each pairwisedisjoint collection of non-empty, convex, open subsets has a choicefunction. van Douwen [1985]. |
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| 118 C | Every linearly orderable space is collectionwisenormal. van Douwen [1985] and Note 71. |
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| 118 D | Every linearly orderable space is collectionwiseHausdorff. van Douwen [1985] and Note 71. |
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| 118 E | Every linearly orderable space satisfies Urysohn'slemma. van Douwen [1985] and Note 141. |
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| 118 F | Every linearly orderable space satisfies theTietze-Urysohn extension theorem. van Douwen [1985] andNote 141. |
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| 118 G | Each complete linear order admits an extreme choice,that is, if \((X,\le)\) is a linear order then there is a choice function \(f\)on the set of non-empty open interval of \(X\) with the property that for all\((a,b,c)\in X^3\) with \(a < b < c\), \(f((a,c))\in\{f((a,b)),b,f((b,c)) \}\).Morillon [1991a]. |
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| 118 H | For each linear order \((X,\le)\), there is an ordinal\(\alpha\) and a strictly increasing injective map \(j\) from \((X,\le)\) to\(\{0,1\}^\alpha\) with the lexicographic order. Morillon [1987]. |
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| 118 I | Every linearly ordered set of linear orders, eachof which is non-empty and conditionally complete, has a choice function.Morillon [1988], van Douwen [1985], and Note 71. |
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| 118 J | If \((X,\le)\) is a linear, complete order, then theorder topology on \(X\) is hereditarily normal. Morillon [1988],van Douwen [1985], and Note 71. |
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| 118 K | If \((X,\le)\) is a linear order, then the ordertopology on \(X\) is highly separated. Morillon [1988] and Note 71. |
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| 118 L | If \(X\) is a linear, complete order, then the ordertopology on \(X\) is highly separated. Morillon [1988] and Note 71. |
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| 118 M | If \((X,\le)\) is a linear order, then the family\(\{A(a,b): a, b\in X \land a< b\}\) has a choice function. (\(A(a,b)\)denotes the set of continuous, order preserving functions from \(X\) into\([0,1]\) such that \(f(a)= 0\) and \(f(b) = 1\).) Morillon [1988]and Note 71. |
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| 118 N | If \((X,\le)\) is a linear order then the family\(\{A'(a,b): a, b\in X \land a< b\}\) has a choice function. (\(A'(a,b)\)denotes the set of continuous functions from \(X\) into \([0,1]\) suchthat \(f(a)= 0\) and \(f(b) = 1\).) Morillon [1988] and Note 71. |
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| 118 O | If \((X,\le)\) is a linear order, then the ordertopology on \(X\) is effectively normal. Morillon [1988] andNote 71. |
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| 118 P | If \((X,\le)\) is a complete linear order, then theorder topology on \(X\) is effectively normal. Morillon [1988]and Note 71. |
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| 118 Q | If \((X,\le)\) is a linear order, then the ordertopology on \(X\) is hereditarily normal. Morillon [1988] andNote 71. |
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| 118 R | If \((X,\le)\) is a linear order, then the ordertopology on \(X\) is monotonely normal. Morillon [1987] andNote 71. |
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| 118 S | If \((X,\le)\) is a complete linear order then there isa choice function \(*\) for the family of non-empty open intervals of \(X\)such that if \(a,b,c\in X\) and either \((a,b)\ne \emptyset\) or\((b,c)\ne \emptyset\) then \(*((a,c)) \in \{ *((a,b)), *((b,c)), b,b^+\}\)(where \(b^+\) is the successor of \(b\) if it exists and \(b\)otherwise). Morillon [1987] and Note 71. |
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| 118 T | Every complete linear order has an increasing choicefunction. Morillon [1987] and Note 71. |
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| 118 U | Every linearly ordered space is monotonically normal.(A space \((X,<)\) is monotonically normal if there is an operator\(V\) which assigns to each \(x\in X\) and basic open neighborhood \(U\) of \(x\)a basic open neighborhood \(V(x,U)\) of \(x\) such that \(V(x,U)\cap V(x',U')\ne\emptyset\) implies \(x\in U'\) or \(x'\in U\).) Good/Tree [1995]. |
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| 118 V | If \((L,\le)\) is a linear order and \(G=\{[l_i,r_i]:i\in K\}\) is a locally finite family of closed intervals of \(L\) whoseinteriors are non-empty and pairwise disjoint, then there exists acontinuous real valued function \(f: L\to \Bbb R\) such that \(|f^{-1}(0)\cap[l_i,r_i]|\ge 2\) and \(|f^{-1}(1)\cap[l_i,r_i]|\ge 1\) for all \(i\in K\).Keremedis [1997] and Note 43. |
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| 118 W | If \((L,\le)\) is a linear order and \(G=\{[l_i,r_i]:i\in K\}\) is a locally finite family of closed intervals of \(L\) whoseinteriors are non-empty and pairwise disjoint, then there exists acontinuous real valued function \(f: L\to \Bbb R\) such that for each\(i\in K\), \(f(l_i) = f(r_i) = 0\) and \(f(x) > 0\) for all \(x\in (l_i, r_i)\).Keremedis [1997] and Note 43. |
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| 118 X | If \((L,\le)\) is a complete linear order and \(G=\{(l_i,r_i): i\in K\}\) is a family of open, non-empty, pairwisedisjoint intervals of \(L\), then there exists a set \(\{f_i: i\in K\}\)of non constant continuous real valued functions on \(L\) such thatfor each \(i\in K\), \(f_i\) vanishes outside of \((l_i,r_i)\). Keremedis [1997]. |
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