# Connectifying a topological space by adding one point

Document Type : Research Article

Author

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

Abstract

P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a point at infinity'') if and only if it is non-compact. Also he asked for characterizations of spaces which have one-point connectifications. Here, we study one-point connectifications, and in an attempt to answer Alexandroff's question (and in analogy with Alexandroff's theorem) we prove that in the class of $T_i$-spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a one-point connectification if and only if it has no compact component. We extend this theorem to the case $i=6$ by assuming the set-theoretic assumption $\mathbf{MA}+\neg\mathbf{CH}$, and to the case $i=2$ by slightly modifying its statement. We further extended the theorem by proving that a locally connected metrizable (resp. paracompact) space has a metrizable (resp. paracompact) one-point connectification if and only if it has no compact component. Contrary to the case of the one-point compactification, a one-point connectification, if exists, may not be unique. We instead consider the collection of all one-point connectifications of a locally connected locally compact space in the class of $T_i$-spaces ($i=3\frac{1}{2},4,5$). We prove that this collection, naturally partially ordered, is a compact conditionally complete lattice whose order structure determines the topology of all Stone--$\rm‎\check{C}$‎ech remainders of components of the space.

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#### References

1. [1] M. Abry, J.J. Dijkstra and J. van Mill, On one-point connectications, Topology Appl. 154 (2007), no. 3, 725--733.

[2] O.T. Alas, M.G. Tkacenko, V.V. Tkachuk and R.G. Wilson, Connectifying some spaces, Topology Appl. 71 (1996),

1. 3, 203--215.

[3] P. Alexandroff, Uber die Metrisation der im Kleinen kompakten topologischen Raume, (German) Math. Ann. 92

(1924), no. 3-4, 294--301.

[4] D.K. Burke, Covering Properties, Handbook of Set-Theoretic Topology, 347-422, North-Holland, Amsterdam, 1984.

[5] W.W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer--Verlag, New York-Heidelberg, 1974.

[6] J.J. Dijkstra and J. van Mill, Erd}os space and homeomorphism groups of manifolds, Mem. Amer. Math. Soc. 208

(2010), no. 979, vi+62 pp.

[7] J.J. Dijkstra, J. van Mill and K.I.S Valkenburg, On nonseparable Erd}os spaces, J. Math. Soc. Japan 60 (2008), no.

3, 793--818.

[8] J.J. Dijkstra and D. Visser, On generalized Erd}os spaces, Topology Appl. 155 (2008), no. 4, 233--251.

[9] J.M. Domnguez, A generating family for the Freudenthal compactification of a class of rimcompact spaces, Fund.

Math. 178 (2003), no. 3, 203--215.

[10] A. Dow and V.V. Tkachuk, Connected compactifications of countable spaces, Houston J. Math. 37 (2011), no. 2,

689--698.

[11] R. Engelking, General Topology, Second edition, Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin,

1989.

[12] L. Gillman and M. Jerison, Rings of Continuous Functions, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-

London-New York 1960.

[13] G. Gruenhage, J. Kulesza and A. Le Donne, Connectifications of metrizable spaces, Topology Appl. 82 (1998), no.

1-3, 171--179.

[14] M. Henriksen, L. Janos and R.G. Woods, Properties of one-point completions of a noncompact metrizable space,

Comment. Math. Univ. Carolin. 46 (2005), no. 1, 105--123.

[15] B. Honari and Y. Bahrampour, Cut-point spaces, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2797--2803.

[16] B. Knaster, Sur un probleme de P. Alexandroff, (French) Fund. Math. 33 (1945) 308--313.

[17] M.R. Koushesh, On one-point metrizable extensions of locally compact metrizable spaces, Topology Appl. 154

(2007), no. 3, 698--721.

[18] M.R. Koushesh, On order structure of the set of one-point Tychonoff extensions of a locally compact space, Topology

Appl. 154 (2007), no. 14, 2607--2634.

[19] M.R. Koushesh, Compactification-like extensions, Dissertationes Math. 476 (2011) 88 pages.

[20] M.R. Koushesh, The partially ordered set of one-point extensions, Topology Appl. 158 (2011), no. 3, 509--532.

[21] M.R. Koushesh, One-point extensions and local topological properties, Bull. Aust. Math. Soc. 88 (2013), no. 1,

12--16.

[22] M.R. Koushesh, Topological extensions with compact remainder, J. Math. Soc. Japan 67 (2015), no. 1, 1--42.

[23] M.R. Koushesh, One-point connectifications, J. Aust. Math. Soc. 99 (2015), no. 1, 76--84.

[24] A. Leiderman and M. Tkachenko, Some properties of one-point extensions, Topology Proc. 59 (2022) 195--208.

[25] J. Mack, M. Rayburn and R.G. Woods, Local topological properties and one point extensions, Canad. J. Math. 24

(1972) 338--348.

[26] J. Mack, M. Rayburn and R.G. Woods, Lattices of topological extensions, Trans. Amer. Math. Soc. 189 (1974)

163--174.

[27] K.D. Magill, Jr. The lattice of compactifications of a locally compact space, Proc. London Math. Soc. (3) 18 (1968)

231--244.

[28] F. Mendivil, Function algebras and the lattice of compactifications, Proc. Amer. Math. Soc. 127 (1999), no. 6,

1863--1871.

[29] S. Mrowka, On local topological properties, Bull. Acad. Polon. Sci. Cl. III 5 (1957) 951--956.

[30] S. Mrowka, On the unions of Q-spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 6 (1958) 365--368.

[31] S. Mrowka, Some comments on the author's example of a non-R-compact space, Bull. Acad. Polon. Sci. Ser. Sci.

Math. Astronom. Phys. 18 (1970), no. 8, 443--448.

[32] A. Mysior, A union of realcompact spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. 29 (1981), no. 3-4, 169--172.

[33] J.R. Porter and C. Votaw, H-closed extensions, II Trans. Amer. Math. Soc. 202 (1975) 193--209.

[34] J.R. Porter and R.G. Woods, Extensions and Absolutes of Hausdorff Spaces. Springer, New York, 1988.

[35] J.R. Porter and R.G. Woods, The poset of perfect irreducible images of a space, Canad. J. Math. 41 (1989), no. 2,

213--233.

[36] J.R. Porter and R.G. Woods, Subspaces of connected spaces, Topology Appl. 68 (1996), no. 2, 113--131.

[37] M.C. Rayburn, On Hausdorff compactifications, Pacific J. Math. 44 (1973) 707--714.

[38] R.M. Stephenson, Jr., Initially κ-compact and related spaces, Handbook of Set-Theoretic Topology, 603--632, North--

Holland, Amsterdam, 1984.

[39] J.E. Vaughan, Countably compact and sequentially compact spaces, Handbook of Set-Theoretic Topology, 569--602,

North--Holland, Amsterdam, 1984.

[40] S. Watson and R.G. Wilson, Embeddings in connected spaces. Houston J. Math. 19 (1993), no. 3, 469--481.

[41] W. Weiss, Countably compact spaces and Martin's axiom, Canad. J. Math. 30 (1978), no. 2, 243--249.

[42] R.G. Woods, Zero-dimensional compactifications of locally compact spaces. Canad. J. Math. 26 (1974) 920--930.