  For a set-valued function $F$ on a compact subset $W$ of a manifold, spanning
is a topological property that implies that $F(x) \ne 0$ for interior points
$x$ of $W$. A myopic equilibrium applies when for each action there is a payoff
whose functional value is not necessarily affine in the strategy space. We show
that if the payoffs satisfy the spanning property, then there exist a myopic
equilibrium (though not necessarily a Nash equilibrium). Furthermore, given a
parametrized collection of games and the spanning property to the structure of
payoffs in that collection, the resulting myopic equilibria and their payoffs
have the spanning property with respect to that parametrization. This is a far
reaching extension of the Kohberg-Mertens Structure Theorem. There are at least
four useful applications, when payoffs are exogenous to a finite game tree (for
example a finitely repeated game followed by an infinitely repeated game), when
one wants to understand a game strategically entirely with behaviour
strategies, when one wants to extends the subgame concept to subsets of a game
tree that are known in common, and for evolutionary game theory. The proofs
involve new topological results asserting that spanning is preserved by
relevant operations on set-valued functions.
