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**A lattice of conditions on topological spaces.**
*(English)*
Zbl 0562.54043

We answer some questions raised in an earlier paper by two of the authors [P. J. Collins and A. W. Roscoe, Proc. Amer. Math. Soc. 90, 631–640 (1984; Zbl 0541.54034)] and to shed further light on the type of condition discussed there. Each condition applies to a topological space \(X\) for each element \(x\) of which a family \(W(x)\) of subsets containing x is given, such that if \(x\in U\) and \(U\) is open, then there exists an open \(V=V(x,U)\) containing \(x\) such that \(x\in W\subseteq U\) for some \(W\in W(y)\) whenever \(y\in V\). Different conditions are obtained by placing restrictions on the structure of the families \(W(x)\). The range of spaces encompassed by these conditions varies from arbitrary topological spaces to those which are metrisable. In particular, new criteria for monotone normality, paracompactness, and for having the property of being a Nagata space are established.

The main result of the paper is the following. Theorem. In order that a \(T_ 1\)-space X be metrizable it is necessary and sufficient that, for each x in X, there be a countable decreasing local basis \(\{W(n,x):n=1,2,\dots\}\) of open sets at \(x\) satisfying if \(x\in U\) and \(U\) is open, then there exists an open \(V=V(x,U)\) containing \(x\) and such that \(x\in W(n,y)\subseteq U\) for some integer \(n=n(x,y,U)\) whenever \(y\in V\).

The main result of the paper is the following. Theorem. In order that a \(T_ 1\)-space X be metrizable it is necessary and sufficient that, for each x in X, there be a countable decreasing local basis \(\{W(n,x):n=1,2,\dots\}\) of open sets at \(x\) satisfying if \(x\in U\) and \(U\) is open, then there exists an open \(V=V(x,U)\) containing \(x\) and such that \(x\in W(n,y)\subseteq U\) for some integer \(n=n(x,y,U)\) whenever \(y\in V\).

Reviewer: Peter J. Collins (Oxford)

### MSC:

54E35 | Metric spaces, metrizability |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |

### Keywords:

semimetric; monotone normality; paracompactness; Nagata space; metrizability; stratifiability; first countability### Citations:

Zbl 0541.54034
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\textit{P. J. Collins} et al., Proc. Am. Math. Soc. 94, 487--496 (1985; Zbl 0562.54043)

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### References:

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