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**1 - 4**of**4**### FUNDAMENTAL PUSHOUT TOPOSES

"... Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spir ..."

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Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.

### © Electronic Publishing House NOTES ON FRÉCHET SPACES

, 1998

"... Abstract. First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures. Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space a ..."

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Abstract. First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures. Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet. Finally, we study on a sufficient condition that a sequential space be Fréchet, a weakly first countable space be first countable, and a symmetrizable space be semi-metrizable.

### Abstract How Do Domains Model Topologies?

"... In this brief study we explicitly match the properties of spaces modelled by domains with the structure of their models. We claim that each property of the modelled topology is coupled with some construct in the model. Examples are pairs: (i) firstcountability- strictly monotone map, (ii) developabi ..."

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In this brief study we explicitly match the properties of spaces modelled by domains with the structure of their models. We claim that each property of the modelled topology is coupled with some construct in the model. Examples are pairs: (i) firstcountability- strictly monotone map, (ii) developability- measurement, (iii) metrizability- partial metric, (iv) ultrametrizability- tree, (v) Choquet-completeness-dcpo, and more. By making this correspondence precise and explicit we reveal how domains model topologies. 1