Topological games and manifolds
by
David Gauld
The University of Auckland

A topological game is usually played on a topological space X and usually involves two players, one of whom may choose a subset of X subject to some rules and the other a point in the set. Then the collection of points chosen may determine who wins the game depending on whether the set satisfies some property or other. As an example we have the following game. Let X be a topological space and U be an open cover of X. At the \alphath step the first player chooses a dense subset D_\alpha subset X then the second player chooses a point x_\alpha in D_\alpha; the second player wins if for some \alpha the set F_\alpha = { x_\beta  / \beta < \alpha} is closed and discrete in X and st(F_\alpha, U)=X.

A topological manifold is a connected Hausdorff topological space each point of which has a neighbourhood homeomorphic to euclidean space Rⁿ. Of course Rⁿ is itself a manifold. So is Sⁿ, the set of all points in Rⁿ⁺¹ whose distance from the origin is 1.

In this talk several topological games will be described and the outcome of playing these games on a manifold will be explored. In particular it is found that the outcome of some games is related to whether the manifold is metrisable.

Date received: June 2, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacc-31.