Separate and Joint Continuity

Jiling Cao is a Professor of Mathematics at Auckland University of Technology. He received his PhD from The University of Auckland in 1999. He has published over 80 research articles in the areas of general topology, functional analysis, mathematical economics, and financial mathematics. He is a Fellow of the New Zealand Mathematics Society and holds visiting professorship positions at several other universities. From 2015 to present, he has been the Head of the Department of Mathematical Science at Auckland University of Technology. Warren B. Moors is a Professor of Mathematics at the University of Auckland. He has published over 85 research articles in the areas of: functional analysis, general topology and optimisation. He is a Fellow of both the Australian Mathematical Society and the New Zealand Mathematical Society and is the recipient of the 2001 New Zealand Mathematical Society Research Award. He received his PhD from the University of Newcastle in 1992.

1. Introduction. 1.1. Background. 1.2. Baire and Related Spaces. 1.3. Quasicontinuous Functions. 1.4. Set-Valued Mappings. 1.5. Basics of Function Spaces. 1.6. Concepts in Banach Spaces. 1.7. Commentary and Exercises. 2. Fundamental Results. 2.1. Fundamental Questions. 2.2. First Countable Spaces. 2.3. q-Spaces. 2.4. Second Countable Spaces. 2.5. Separately Quasicontinuous Functions. 6. Piotrowski’s Theorem. 2.7. Talagrand’s Problem. 2.8. Commentary and Exercises. 3. Continuity of Group Actions and Operations. 3.1. Semitopological and Paratopological Groups. 3.2. Δ-Baire Spaces. 3.3. Continuity of Group Actions. 3.4. Some Counterexamples. 3.5. Miscellaneous Applications. 3.6. Commentary and Exercises. 4. Namioka Theorem and Related Spaces. 4.1. Namioka Theorem. 4.2. Namioka Theorem - a Functional Analytic Proof. 4.3. Namioka Spaces. 4.4. Co-Namioka Spaces. 4.5. Commentary and Exercises. 5. Various Applications. 5.1. Point of Continuity Properties. 5.2. Minimal USCO Mappings. 5.3. Ryll-Nardzewski Fixed-Point Theorem. 5.4. Differentiability of Continuous Convex Functions. 5.5. Applications in Variational Analysis. 6. Future Directions and Open Problems. 6.1. Topologies of Separate and Joint Continuity. 6.2. Semitopological and Paratopological Groups. 6.3. Namioka Spaces. 6.4. Co-Namioka and Related Spaces. 6.5. Baire Measurability of Separately Continuous Functions. 6.6. Sets of Discontinuity Points of Separately Continuous Functions. 6.7. Various Maslyuchenko Spaces.