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The class of sequential spaces and its successive smaller subclasses, the Fréchet spaces and the first-countable spaces, have topologies which are completely specified by their convergent sequences. Because sequences have many advantages over nets, these topological spaces are of interest. Special attention is paid to those properties of first-countable spaces which can or cannot be generalized to Fréchet or sequential spaces. For example, countable compactness and sequential compactness are equivalent in the larger class of sequential spaces. On the other hand, a Fréchet space with unique sequential limits need not be Hausdorff, and there is a product of two Fréchet spaces which is not sequential. Some of the more difficult problems are connected with products. The topological product of an arbitrary sequential space and a T₃ (regular and T₁) sequential space X is sequential if and only if X is locally countably compact. There are also several results which demonstrate the non-productive nature of Fréchet spaces. The sequential spaces and the Fréchet spaces are precisely the quotients and continuous pseudo-open images, respectively, of either (ordered) metric spaces or (ordered) first-countable spaces. These characterizations follow from those of the generalized sequential spaces and the generalized Fréchet spaces. The notions of convergence subbasis and convergence basis play an important role here. Quotient spaces are characterized in terms of convergence subbases, and continuous pseudo-open images in terms of convergence bases. The equivalence of hereditarily quotient maps The class of sequential spaces and its successive smaller subclasses, the Fréchet spaces and the first-countable spaces, have topologies which are completely specified by their convergent sequences. Because sequences have many advantages over nets, these topological spaces are of interest. Special attention is paid to those properties of first-countable spaces which can or cannot be generalized to Fréchet or sequential spaces. For example, countable compactness and sequential compactness are equivalent in the larger class of sequential spaces. On the other hand, a Fréchet space with unique sequential limits need not be Hausdorff, and there is a product of two Fréchet spaces which is not sequential. Some of the more difficult problems are connected with products. The topological product of an arbitrary sequential space and a T₃ (regular and T₁) sequential space X is sequential if and only if X is locally countably compact. There are also several results which demonstrate the non-productive nature of Fréchet spaces. The sequential spaces and the Fréchet spaces are precisely the quotients and continuous pseudo-open images, respectively, of either (ordered) metric spaces or (ordered) first-countable spaces. These characterizations follow from those of the generalized sequential spaces and the generalized Fréchet spaces. The notions of convergence subbasis and convergence basis play an important role here. Quotient spaces are characterized in terms of conver-gence subbases, and continuous pseudo-open images in terms of convergence bases. The equivalence of hereditarily quotient maps and continuous pseudo-open maps implies the latter result. and continuous pseudo-open maps implies the latter result. / Science, Faculty of / Mathematics, Department of / Graduate

A topological space may be viewed as an algebraic structure. For example, it may be viewed as a (complete atomic) Boolean algebra equipped with a closure operator. The lattice of closed subsets is another algebraic structure which may be associated with a topological space. Tne purpose of this thesis is primarily to investigate the metamathematical properties of algebraic structures associated with topological spaces. More specifically, we will first consider questions of decidability of the theories of these algebraic structures. It turns out that these theories are undecidable. We will also examine certain equivalence relations on the class of topological spaces that arise naturally from viewing them as first-order structures. Finally we will show that certain classical theorems of model theory do not hold for topological spaces. / Science, Faculty of / Mathematics, Department of / Graduate

Bibliography: pages 96-103. / In the early 1940's, A.D. Alexandroff [1940), [1941) and [1943] introduced a concept of space, more general than topological space, in order to obtain a simple connection between a space and the system of real-valued functions defined on it. Such a connection aided the investigation of the relationships between the linear functionals on these systems of functions and the additive set functions defined on the space. The Alexandroff spaces of this thesis are what Alexandroff himself called the completely normal spaces and what H. Gordon [1971) called the zero-set spaces.

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The purpose of the paper is to prove that the Lp spaces, p ≥ 1, of equivalence classes of functions are Banach spaces.

A Dissertation Submitted of the Faculty of Science, University of the Witwatersrand, Johannesburg in Partial Fulfilment of the Requirements for the Degree of Master of Science / Let 4 be the class of pairs of loc ..My onvex spaces (X,V) “h ‘ch are such that every closed graph linear ,pp, 1 from X into V is continuous. It B is any class of locally . ivex l.ausdortf spaces. let & w . (X . (X.Y) e 4 for ,11 Y E B). " ‘his expository dissertation, * (B) is investigated, firstly i r arbitrary B . secondly when B is the class of C,-complete paces and thirdly whon B is a class of locally convex webbed s- .ces

A continuous convex function on an open interval of the real line is differentiable everywhere except on a countable subset of its domain. There has been interest in the problem of characterizing those Banach spaces where the continuous functions exhibit similar differentiability properties. In this paper we show that if a Banach space E has property (H*) and B_E• is weak* sequentially compact, then E is an Asplund space. In the case where the space is weakly compactly generated, it is shown that property (H*) is equivalent for the space to admit an equivalent Frechet differentiable norm. Moreover, we define the SH* spaces, show that every SH* space is an Asplund space, and show that every weakly sequentially complete SH* space is reflexive. Also, we study the relation between property (H*) and the asymptotic norming property (ANP). By a slight modification of the ANP we define the ANP*, and show that if the dual of a Banach spaces has the ANP*-I then the space admits an equivalent Fréchet differentiability norm, and that the ANP*-II is equivalent to the space having property (H*) and the closed unit ball of the dual is weak* sequentially compact. Also, we show that in the dual of a weakly countably determined Banach space all the ANP-K'S are equivalent, and they are equivalent for the predual to have property (H*).