Relations between pairs of separation axioms are considered. Given two separation axioms, it is investigated whether or not a topological space having the property of one of the separation axioms has the property of the other. Eighteen separation axioms are considered and the relation between the members of pairs of separation axioms is determined in every possible case. That is, with each pair of separation axioms there is associated a theorem showing the relative strengths of the members or an example showing their relative independence. As a secondary interest, some characterizations of most of the eighteen separation axioms are given. Also some necessary conditions for normal spaces and completely normal spaces are generalized. / Science, Faculty of / Mathematics, Department of / Graduate

The problem of this paper is to define a category of topological spaces and sheaves in topological and algebraic terms. It is then necessary to show that if the topological space is, in particular, affine space and the sheaf over it the sheaf of germs of regular functions* then the pair, consisting of the topological space and the sheaf of germs of regular functions, called an affine variety, is an element of the category. We then generalize this result by showing that algebraic varieties belong to the category. Before defining the category, it is necessary to establish in Section 1 some elementary results of general topology which are used in proving properties of the category. It is also necessary to define the Zariski topology in purely topological terms. This is done in Section 2. / Science, Faculty of / Mathematics, Department of / Graduate

Given a Radón measure on each of two locally compact Hausdorff spaces we consider three product measures on the product space - the classical product measure generated by measurable rectangles as in Munroe, the product measure generated by measurable rectangles and nil sets of Bledsoe and Morse, and a Radón product measure obtained by a Riesz Representation Theorem from the product linear functional of Bourbaki. For the classical product measure we prove a Fubini Theorem for compact sets although we do not know whether compact sets are measurable, a theorem giving necessary and sufficient conditions for open sets to be measurable, and a theorem that every closed СɣϬ is measurable. We prove that all three product measures agree on compact sets and thus that the Bledsoe-Morse product measure and the Radón product measure agree on open sets. Finally, we give examples to show that certain results cannot be extended. / Science, Faculty of / Mathematics, Department of / Graduate

In their monograph 'Quasi-uniform Topological Spaces’, M.G-. Murdeshwar and S.A. Naimpally documented those "uniformity" results which carry over to quasi-uniformities, a quasi-uniformity being a uniformity that lacks the symmetry property. It seemed natural to ask what results might remain true if the "triangle" property of a uniformity were also removed. The investigation of this idea here has resulted in a rather primitive (from a topological point of view) structure called a "diagonal space". Unfortunately, since a topology can not be obtained in the usual way from such a diagonal structure, most of the desirable standard results do not carry over. In Chapter 0 the basic notions that are needed are defined, the most important being that of a filter. Chapter 1 deals with diagonal spaces and the pseudo-topologies that they generate. The latter part of Chapter 1 outlines techniques whereby a pseudo-topology can be "reduced" to a topology. The relationship between diagonal filters and the "pretopologies" of D.C. Kent is discussed in Chapter 2 along with the various relationships between the topologies and generalizations of topolgies that can be defined in a natural way from diagonal spaces and pretopologies. Finally, in Chapter 3, there is a very brief discussion on the analogues, in terms of diagonal spaces and pretopologies, of a few standard concepts of topology. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis we construct some examples of thick subalgebras ɛ of factors ɑ. ɛ is thick in ɑ if (ɛ' ∩ ɑ) is maximal abelian in ɑ. We are concerned with their inner equivalence: given the thick subalgebras ɛ and ℱ in ɑ, does there exist a unitary U є ɑ such that U є U* = ℱ ? Examples of thick subalgebras which are not maximal abelian have been given by Dixmier and Kadison. Later Bures constructed numerous examples which he distinguished by use of certain invariants. We use Bures's construction to get, in certain factors ɑ of types II₁,- II₀₀, III, uncountable families {ɛ[subscript i]: iєℱ} of thick subalgebras of ɑ such that ɛ[subscript i] is not inner equivalent to ɛ[subscript J] when i ≠ J (We are able to add one example to those constructed by Bures). In each family, the ɛ[subscript i]cannot be distinguished by means of Bures's invariants, and so we are forced to show their non-inner-equivalence by direct calculations. / Science, Faculty of / Mathematics, Department of / Graduate

Let S be a topological semigroup and C(S) the space of bounded real continuous function on S with sup norm. For f є C(S), s, t є S, [subscript s]f(t) = f(st), let LUC(S) be the space of all f є C(S) for which the map s ↦ [subscript s] f from S to C(S) is continuous, Δ(S) the set of multiplicative means on LUC(S), and CoΔ(S) the convex hull of Δ(S). In this thesis we study and characterize topological semigroup S for which (*) LUC(S) has a LIM (left invariant mean) in CoΔ(S). A decomposition theorem for such semigroups has been obtained. We also consider properties that arise from the action of semigroups satisfying (*) on certain topological spaces. In particular, we generalise Mitchell's fixed point theorem (theorem 1 [28]). Other characterization theorems and combinatorial properties for such semigroups have also been obtained. Continuing the work of J. Sorenson [32], we obtain characterizations and functional analytic properties for discrete semigroups satisfying (*), generalising some of the results of Granirer [12], [13], [l14] and Mitchell [26] for semigroups admitting a multiplicative LIM. Finally we characterize all semigroups S for which m(S) (the space of hounded real functions) has a non-trivial translation invariant subalgebra, containing constants and admits a multiplicative LIM. We also give a method, utilizing the class of left thick subsets of Mitchell [25], in constructing a huge class of such subalgebras. Furthermore, we show that the above method and characterization is valid even for semigroup of transformations. Other diverse results in this direction are also obtained. / Science, Faculty of / Mathematics, Department of / Graduate

This paper is an introduction to the study of the problem: what decomposition spaces of Eⁿ are homeomorphic to Eⁿ? An introductory section on the elementary properties of decompositions and decomposition spaces is followed by characterizations of the decomposition spaces of E¹ and E² which are homeomorphic to E¹ and E² respectively. The characterization for E² follows Moore-Whyburn. We then give sufficient conditions for a decomposition space of Eⁿ to be Eⁿ, following Bing's recent work. No characterization is known for general Eⁿ . This paper reflects the status of the question as of, roughly, 1957. There has been essentially no progress since then. / Science, Faculty of / Mathematics, Department of / Graduate

The strict topology β was first defined on the space of bounded complex-valued continuous functions Cb(X), on a locally compact Hausdorff space X, by Buck. It was found to have many applications in Approximation Theory, spectral synthesis, spaces of bounded holomorphic functions and multipliers of Banach Algebras.

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It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then observe that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we delve into algebraic topology to determine that the aforementioned surfaces are not homeomorphic to one another, and thus we can place each surface into exactly one of these equivalence classes.