Compactifications as closures of graphs

Steiner, A. K.; Steiner, E. F.

doi:10.4064/fm-63-2-221-223

“…We also show how this method of construction relates to one developed by Steiner and Steiner [5] and to Whyburn's Unified Space [6].…”

mentioning

confidence: 99%

Generalizing the Alexandroff-Urysohn Double Circumference Construction

Chandler¹,

Faulkner²,

Guglielmi³

et al. 1981

Proceedings of the American Mathematical Society

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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.

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“…This theory was extended by Rogers [14] and Chandler [3, 7.8] to conclude that if X is locally compact and not pseudocompact then any compact space containing a dense continuous image of R is a remainder of X. One of the useful results in this area has been a theorem of Steiner and Steiner [15]. Here we generalize this theorem and use it to obtain some new results on remainders.…”

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confidence: 80%

Remainders in Hausdorff compactifications

Chandler¹,

Tzung²

1978

Proc. Amer. Math. Soc.

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Abstract. We generalize a theorem of Steiner and Steiner and use it to obtain new results concerning remainders for completely regular spaces. Both the locally compact and the nonlocally compact cases are considered.Introduction. The nature of what one added to a completely regular space in order to compactify it has been considered almost from the beginning of the theory. For example, Cech [2] was concerned about the cardinality of /?N\N. It was not until the mid-1960's, however, that the topological properties of remainders was studied in a concerted fashion. Magill, in a series of papers [8], [9], [10] discovered and developed conditions on a space X which would guarantee that members of a certain class of continua would be remainders of X. This theory was extended by Rogers [14] and Chandler [3, 7.8] to conclude that if X is locally compact and not pseudocompact then any compact space containing a dense continuous image of R is a remainder of X. One of the useful results in this area has been a theorem of Steiner and Steiner [15]. Here we generalize this theorem and use it to obtain some new results on remainders. Throughout we follow the general terminology of [3]. In particular, all spaces will be assumed to be completely regular and Hausdorff. A remainder of X is any aX \ X where aX is a compactification

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“…We also show how this method of construction relates to one developed by Steiner and Steiner [5] and to Whyburn's Unified Space [6].…”

mentioning

confidence: 99%

“…Let A* = X u {w} denote the 1-point compactification of A and let N(u) denote a neighborhood of w. Steiner and Steiner [5] showed that if A is locally compact, K is compact, and if there is a continuous map f: X -* K such that f[N(u) n A] is dense for each neighborhood N(u>), of a, then X has a compactification X with a remainder homeomorphic to K. The closure of the graph of / in X* X K was shown to be such an X.…”

mentioning

confidence: 99%

Generalizing the Alexandroff-Urysohn double circumference construction

Chandler¹,

Faulkner²,

Guglielmi³

et al. 1981

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

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Compactifications as closures of graphs

Cited by 17 publications

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Generalizing the Alexandroff-Urysohn Double Circumference Construction

Generalizing the Alexandroff-Urysohn Double Circumference Construction

Remainders in Hausdorff compactifications

Generalizing the Alexandroff-Urysohn double circumference construction

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