Spaces whose regular preimages are normal

Call a regular space X Pin(κ) if every ⩽ κ-to-1 continuous regular preimage of X is normal. We investigate Pin(κ) for κ ⩽ ω. Some of our results include: Pin(1) spaces are hereditarily normal, every subset of a Pin(ω) space is a Gδ, and there is a ZFC example of a Pin(1) space that is not Pin(2).