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The strongly attached point topology of the abstract boundary for space-time. (English) Zbl 1297.83027

Summary: The abstract boundary construction of Scott and Szekeres provides a ‘boundary’ for any n-dimensional, paracompact, connected, Hausdorff, \({C}^{\infty}\) manifold. Singularities may then be defined as objects within this boundary. In a previous paper [R. A. Barry and S. M. Scott, Classical Quantum Gravity 28, No. 16, Article ID 165003, 14 p. (2011; Zbl 1226.83005)], a topology referred to as the attached point topology was defined for a manifold and its abstract boundary, thereby providing us with a description of how the abstract boundary is related to the underlying manifold. In this paper, a second topology, referred to as the strongly attached point topology, is presented for the abstract boundary construction. Whereas the abstract boundary was effectively disconnected from the manifold in the attached point topology, it is very much connected in the strongly attached point topology. A number of other interesting properties of the strongly attached point topology are considered, each of which support the idea that it is a very natural and appropriate topology for a manifold and its abstract boundary.

MSC:

83C75 Space-time singularities, cosmic censorship, etc.
57R40 Embeddings in differential topology
54F65 Topological characterizations of particular spaces
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics

Citations:

Zbl 1226.83005
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