Yes, I had forgotten: F to be continuous and Y (X and) to be Hausdorff. :)quasar987 said:You need F to be continuous and Y to be Hausdorff and compactly generated. See Corolarry 4.97 of Lee's Introduction to topological manifolds.
A proper homeomorphism is a type of mathematical function that preserves the topological structure of a space. It is a continuous function that has a continuous inverse and maps open sets to open sets.
A proper homeomorphism has the following properties:
A proper homeomorphism is a specific type of homeomorphism. While both are continuous functions with continuous inverses, a proper homeomorphism also has the property of being proper, meaning it maps compact sets to compact sets.
Proper homeomorphisms are useful in many areas of mathematics, including topology, geometry, and functional analysis. They are often used to define equivalence relations between spaces and to study the structure and properties of various mathematical objects.
Yes, the concept of properness can be applied to other types of functions, such as proper mappings or proper embeddings. These functions also preserve the topological structure of a space and have properties similar to proper homeomorphisms.