Page 59 - Textos de Matemática Vol. 34

MINIMAL UNIVERSALITY AND MICHAELLIKE SPACES
SHEILA CARTER AND F. J. CRAVEIRO DE CARVALHO
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. Let X be a topological space and Y be a non-empty subset of X. We can define a topology for X by taking an open set to be the union of an open set of X with a subset of Y . The new topological space is what we will call a Michaellike space, after E. A. Michael who introduced the Michael line [4].
On the other hand a topological space X is universal if any topological space is homeomorphic to a subspace of some topological power of X.
We will show that some minimal, in a sense to be clarified below, universal spaces are quotients of Michaellike spaces obtained from the standard topolo- gical real line.
1. Definition and Notation
Definition 1.1. Let Z be a topological space and Y be a non-empty subset of Z. The Michaellike space ZY has Z as underlying set and each open set is the union ofasubsetofY withanopensetofZ.
If G is a group of homeomorphisms of a topological space Z which acts on it in the obvious way we will denote the quotient space by Z/G and p will denote the projection. For a Michaellike space ZY we will write pY for the projection.
Below Z will be the standard topological R.
2. Discreteness
Theorem 2.1. Let Y be a non-empty subset of R and consider RY . Let G be formedbythehomeomorphismsf:RY →RY suchthatf(Y)=Y.ThenRY/Gis a discrete 2-point space if and only if Y is closed in R.
2000 Mathematics Subject Classification. 54F65.
Key words and phrases. Universal space, Michaellike space, homeomorphism group.
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