Topologies on types: connections

For different purposes, economists may use different topologies on types. We characterize the relationship among these various topologies. First, we show that for any general types, convergence in the uniform-weak topology implies convergence in both the strategic topology and the uniform strategic topology. Second, we explicitly construct a type which is not the limit of any finite types under the uniform strategic topology, showing that the uniform strategic topology is strictly finer than the strategic topology. With these results, we can linearly rank various topologies on the universal type space, which gives a clear picture of the relationship between the implication of types for beliefs and their implication for behaviors. -- the universal type space ; the strategic topology ; the uniform strategic topology ; the uniform-weak topology ; interim correlated rationalizable actions