By Laurent Schwartz
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Extra resources for Radon measures on arbitrary topological spaces and cylindrical measures
Example text
A cover U of the bitopological space (X, τ1 , τ2 ) is defined to be pairwise open if U ⊂ τ1 ∪ τ2 and U contains at least one non-empty member of τ1 and at least one non-empty member of τ2 . If each pairwise open cover of (X, τ1 , τ2 ) has a finite subcover then the space (X, τ1 , τ2 ) is defined to be pairwise compact. Note that (R, U, L) is pairwise compact. Cooke and Reilly [2] considered alternative definitions and characterizations of bitopological compactness. Salbany [13] has provided the most comprehensive early discussion of this topic, based on the stronger definition that (X, τ1 , τ2 ) is pairwise compact if the topological space (X, τ1 ∨ τ2 ) is compact.
Math. 36 (1971), 821–828. Aisling E. McCluskey NUI, Galway, Ireland This Page Intentionally Left Blank B: Basic constructions b-01 Subspaces (hereditary (P)-spaces) b-02 Relative properties b-03 Product spaces b-04 Quotient spaces and decompositions b-05 Adjunction spaces b-06 Hyperspaces b-07 Cleavable (splittable) spaces b-08 Inverse systems and direct systems b-09 Covering properties 60 b-10 Locally (P )-spaces 65 b-11 Rim(P)-spaces b-12 Categorical topology b-13 Special spaces 31 33 37 43 47 49 67 76 74 53 56 This Page Intentionally Left Blank b-1 Subspaces (hereditary (P)-spaces) b-1 31 Subspaces (Hereditary (P)-Spaces) Let X be a given topological space with a topology O (the collection of open subsets) and X a subset of X.
6] H. Fischer, Limesräume, Math. Ann. 137 (1959), 269– 303. [7] M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1–74. [8] W. Gähler, Grundstrukturen der Analysis, AkademieVerlag (1977). [9] H. Hahn, Theorie der reellen Functionen, Berlin (1921). [10] F. Hausdorff, Grundzüge der Mengenlehre, Veit & Comp. (1914). [11] F. Hausdorff, Gestufte Räume, Fund. Math. 25 (1935), 486–506. C. Kent and G. Richardson, Convergence spaces and diagonal conditions, Topology Appl.