Mild Locally Small Spaces

We prove that a direct sum of locally small spaces, each of which is either paracompact or has its admissible open sets closed under arbitrary unions, is mild (all its subspaces are strict) and that mildness is hereditary. We show how to produce examples of non-mild spaces (answering an earlier problem) and consider finitary locally small spaces. We apply the results to paracompact locally definable spaces over o-minimal structures.