Absolutes of Topological Spaces and Their Continuous Maps

Abstract

Dissolution studies were carried out on NaCl crystals, some pure and some doped with various impurities. It was observed that etch pits associated with beaks are obtained only in NaCl crystals doped with CaC12, if the etchant reported by Hari Babu and Bansigir is used. Of the 26 poisons studied with methyl alcohol as base, only a few of the cadmium and lead salts are capable of giving beaks in NaCI:CaCh, whereas with the other poisons etch pits without beaks are obtained. These investigations reveal that the poison, the solvent and the impurity decorating the dislocation line influence the formation of beaks in NaCl crystals. The substructure of the grain boundaries in NaCl:CaC12 crystals has been revealed for the first time, by employing the etching technique. Different types of grain boundaries, junctions between the boundaries and dislocation networks have been recorded.

The essential cover and the absolute cover of a schematic space

“…In [24] the first author has shown that in the category Top of topological spaces with continuous maps, every space X admits a cover p : P → X with respect to the full subcategory Ed of extremally disconnected spaces, and that p coincides with the absolute [23,25,27] of X. To distinguish p from the object P , we will also call P the absolute, and refer to p as the absolute cover of X.…”

“…In [24], the first author proves that every topological space X admits a cover p : P → X with respect to the full subcategory of extremally disconnected spaces, and that p coincides with the absolute [23,25,27] of X (see Section 5). Furthermore, he characterizes the strongly projectable hull in terms of spectra.…”

The Absolute of a Topological Space and its Application to Abelian l-groups

“…We define an absolute (cf. [31,35,38]) of X to be an essentially dense perfect map p : P → X such that P is extremally disconnected. Let Ed ⊂ Top denote the full subcategory of extremally disconnected spaces.…”

“…Apart from being perfect, an absolute p has to be surjective and "irreducible", where the latter concept has found several interpretations or replacements [3,4,29,31]. Our approach is categorical: Since perfect maps are closed, it makes no difference if "surjective" is replaced by dense, i.e.…”

Topological measure theory for completely regular spaces and their projective covers

“…The construction of the absolute has been extended to successively larger classes of spaces by Gleason [12], Strauss [38], and the Russian school [31]. Some of the deepest results on extremally disconnected spaces can be found in Efimov [8].…”