Assume a topological space X is T1. We call X okay if for any two closed subsets A and B, there exists an open set U such that [itex]B \subset U[/itex] and [itex]A \cap U = \emptyset[/itex].
A separation axiom is a property of a topological space that determines how well its points can be separated from each other by different types of sets. It is used to classify and compare different topological spaces.
There are several types of separation axioms, including the most commonly used ones: T0, T1, T2 (also known as Hausdorff), T3, and T4. Each of these axioms specifies different levels of separation between points in a topological space.
The new kind of separation axiom is called T5 and it is an extension of the T4 axiom. It is also known as the completely normal separation axiom. T5 spaces are the most highly separated topological spaces and they have many useful properties that make them important in certain areas of mathematics.
The T5 axiom provides a stronger notion of separation than the previous axioms, allowing for more precise and powerful results in areas such as topology, geometry, and analysis. It also has applications in fields like functional analysis, algebraic topology, and differential geometry.
Yes, there are many examples of T5 spaces, including metric spaces, regular spaces, and even some topological spaces that are not T4. Some common examples of T5 spaces include the real line with the Euclidean topology, the Sorgenfrey line, and the space of continuous functions on a compact Hausdorff space.