Toronto space

In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality.

There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.[1]

The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete.[2]

References

Bonnet, Robert (1993), "On superatomic Boolean algebras", Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Dordrecht: Kluwer Acad. Publ., pp. 31–62, MR 1261195.
van Mill, J.; Reed, George M. (1990), Open problems in topology, Volume 1, North-Holland, p. 15, ISBN 9780444887689.

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[1] Bonnet, Robert (1993), "On superatomic Boolean algebras", Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Dordrecht: Kluwer Acad. Publ., pp. 31–62, MR 1261195.

[2] van Mill, J.; Reed, George M. (1990), Open problems in topology, Volume 1, North-Holland, p. 15, ISBN 9780444887689.