A New Approach to the Katětov–Tong Theorem

Abstract

We give a new proof of the Katětov–Tong theorem. Our strategy is to first prove the theorem for compact Hausdorff spaces, and then extend it to all normal spaces by showing how to extend upper and lower semicontinuous real-valued functions to the Stone–Čech compactification so that the less than or equal relation between the functions is preserved. In this way, the main step of the proof is the compact case, and our approach to handling this case also leads to a new proof of a version of the Stone–Weierstrass theorem.

MSC::

• Primary 54D15
• Secondary 54C30
• 54D30
• 54D35

Acknowledgment

The authors thank the referees for their careful reading of the manuscript and for their comments which have improved the presentation of the paper.

Additional information

Notes on contributors

Guram Bezhanishvili

Guram Bezhanishvili received his Ph.D. from Tokyo Institute of Technology in 1998. In 2000 he joined the faculty at New Mexico State University, where he is currently professor of mathematics. His research is in topology, algebra, and categories in logic (TACL).

Patrick J. Morandi

Patrick Morandi is professor emeritus at New Mexico State University. He continues to conduct mathematical research in lattice-ordered rings, ordered topological spaces, and pointfree topology. He also works with K-12 public school teachers as part of the Mathematically Connected Communities project. His non-academic interests include tennis, golf, cooking, and especially playing with his dogs.

Bruce Olberding

Bruce Olberding received his Ph.D. from Wesleyan University in 1996. In 2002 he joined the faculty at New Mexico State University, where he is currently professor of mathematics. His research interests include commutative algebra, valuation theory, and topology.