separation axiom

For more on this issue, see History of the separation axioms.

Some of these restrictions are given by the separation axioms.

The proof relies on the use of the full separation axiom.

The degree to which any two points can be separated is specified by the separation axioms.

The T1 and R0 properties are examples of separation axioms.

This condition, called the T condition, is one of the separation axioms.

This is how the T axiom fits in with the rest of the separation axioms.

In fact, uniformizability is equivalent to a common separation axiom:

A G space may thus be regarded as a space satisfying a different kind of separation axiom.

In general, quotient spaces are ill-behaved with respect to separation axioms.

When applied to the separation axioms, this leads to the relationships in the table below:

Examples of such properties include connectedness, compactness, and various separation axioms.

Like all separation axioms, completely regularity is not preserved by taking final topologies.

These conditions are examples of separation axioms.

Compactness conditions together with preregularity often imply stronger separation axioms.

In terms of separation axioms, R is a perfectly normal Hausdorff space.

For a detailed treatment, see separation axiom.

Some of these terms are defined differently in older mathematical literature; see history of the separation axioms.

The partition topology provides an important example of the independence of various separation axioms.

The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points.

In fact, fully normal spaces actually have more to do with paracompactness than with the usual separation axioms.

(This condition is known as the Priestley separation axiom.)

It can also be considered as a theory in second-order logic, where now the separation axiom is just a single axiom.

General discussion of the various separation axioms is in the article Separation axiom.

However, the term "separation axiom" has stuck.