A scalar delay-differential equation with diffusion term in one space dimension, where the diffusivity D is a bifurcation parameter, is considered. The center manifold theory and the method of Lyapunov-Schmidt are used to describe two bifurcations from spatially constant solutions as D decreases. By modifying the equation the order of these bifurcations can be reversed. Then the existence of a compact attractor for a class of such equations is shown and the structure of part of the attractor for the modified equation is investigated. It is known that the solutions are globally L2-bounded; bounds on the solution operator from one intermediate space to another are constructed to obtain an attractor in the W e'2 sense.

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Abstract. This paper is a unification of the "Alexandroff-Urysohn Double Circumference Construction", the one-point compactification, the Steiner and Steiner remainder theorem, and Whyburn's Unified Space. All of these are shown to be different aspects of a single construction.The celebrated double circumference of Alexandroff and Urysohn was originally described in [1, pp. 13-15]. It is constructed by taking the union of two concentric circles and choosing the topology on the outer circle to be discrete. On the inner circle a basic neighborhood of a point x is defined to be the union of an open arc on the inner circle containing x and the corresponding (by radial projection) open arc on the outer circle with the point which corresponds to x deleted.Engelking [4] generalized this construction by defining a similar topology on the union of two sets in the case where the first has a compact topology and the second is the same set but with the discrete topology.In the examples above, the spaces constructed are compact Hausdorff spaces each of which contains a discrete set as a dense subset. From our point of view the examples provide us with a means of obtaining compactifications of these discrete spaces. Recall that a compactification of a locally compact space A" is a compact Hausdorff space Z such that X is embedded densely in Z. Z \ X is then called a remainder of X.In this paper we develop a method of compactification which generalizes the above examples. We also show how this method of construction relates to one developed by Steiner and Steiner [5] and to Whyburn's Unified Space [6].We would like this method to give a compactification of an arbitrary (not necessarily discrete) locally compact space. We first note that both the previous constructions require a correspondence between points in the space X which was compactified and points in the remainder Y. In our generalization this correspondence will be given by a continuous map/: X -» Y.Mimicking Engelking we describe the topology on X u Y as follows: For a point p in X, basic neighborhoods in X u Y will be the neighborhoods of p in the original topology on X. For a point q in Y and U an original neighborhood of q in Y we start with/"'(i/) u U and make the following observations.

Abstract. This paper is a unification of the "Alexandroff-Urysohn Double Circumference Construction", the one-point compactification, the Steiner and Steiner remainder theorem, and Whyburn's Unified Space. All of these are shown to be different aspects of a single construction.The celebrated double circumference of Alexandroff and Urysohn was originally described in [1, pp. 13-15]. It is constructed by taking the union of two concentric circles and choosing the topology on the outer circle to be discrete. On the inner circle a basic neighborhood of a point x is defined to be the union of an open arc on the inner circle containing x and the corresponding (by radial projection) open arc on the outer circle with the point which corresponds to x deleted.Engelking [4] generalized this construction by defining a similar topology on the union of two sets in the case where the first has a compact topology and the second is the same set but with the discrete topology.In the examples above, the spaces constructed are compact Hausdorff spaces each of which contains a discrete set as a dense subset. From our point of view the examples provide us with a means of obtaining compactifications of these discrete spaces. Recall that a compactification of a locally compact space A" is a compact Hausdorff space Z such that X is embedded densely in Z. Z \ X is then called a remainder of X.In this paper we develop a method of compactification which generalizes the above examples. We also show how this method of construction relates to one developed by Steiner and Steiner [5] and to Whyburn's Unified Space [6].We would like this method to give a compactification of an arbitrary (not necessarily discrete) locally compact space. We first note that both the previous constructions require a correspondence between points in the space X which was compactified and points in the remainder Y. In our generalization this correspondence will be given by a continuous map/: X -» Y.Mimicking Engelking we describe the topology on X u Y as follows: For a point p in X, basic neighborhoods in X u Y will be the neighborhoods of p in the original topology on X. For a point q in Y and U an original neighborhood of q in Y we start with/"'(i/) u U and make the following observations.

No abstract