In mathematics, in the realm of topology, a paranormal space (Nyikos 1984) is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion.

• Collectionwise normal space – Property of topological spaces stronger than normality
• Locally normal space
• Monotonically normal space – Property of topological spaces stronger than normality
• Normal space – Type of topological space – a topological space in which every two disjoint closed sets have disjoint open neighborhoods
• Paracompact space – Topological space in which every open cover has an open refinement that is locally finite – a topological space in which every open cover admits an open locally finite refinement
• Separation axiom – Axioms in topology defining notions of "separation"

• Nyikos (1984), "Problem Section: Problem B. 25", Top. Proc., 9
• Smith, Kerry D.; Szeptycki, Paul J. (2000), "Paranormal spaces under ◊*", Proceedings of the American Mathematical Society, 128 (3): 903–908, doi:10.1090/S0002-9939-99-05032-7, ISSN 0002-9939, MR 1622981