History of the separation axioms
Encyclopedia
In general topology
, the separation axiom
s have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept.
, there were many definitions offered, some of which assumed (what we now think of as) some separation axioms. For example, the definition given by Felix Hausdorff
in 1914 is equivalent to the modern definition plus the Hausdorff separation axiom
.
The separation axioms, as a group, became important in the study of metrisability: the question of which topological spaces can be given the structure of a metric space
. Metric spaces satisfy all of the separation axioms; but in fact, studying spaces that satisfy only some axioms helps build up to the notion of full metrisability.
The separation axioms that were first studied together in this way were the axioms for accessible spaces, Hausdorff space
s, regular space
s, and normal space
s. Topologists assigned these classes of spaces the names T1, T2, T3, and T4. Later this system of numbering was extended to include T0, T2½, T3½ (or Tπ), and T5.
But this sequence had its problems. The idea was supposed to be that every Ti space is a special kind of Tj space if i > j. But this is not necessarily true. It depended on precisely how the definitions were phrased. For example, a regular space (called T3) does not have to be a Hausdorff space (called T2), at least not according to the simplest definition of regular spaces.
Topologists working on the metrisation problem generally did assume T1; after all, all metric spaces are T1. Thus, they used the simplest definitions for the Ti. Then, for those occasions when they did not assume T1, they used words ("regular" and "normal") for the more complicated definitions, in order to contrast them with the simpler ones. This approach reached its zenith in 1970 with the publication of Counterexamples in Topology
by Lynn A. Steen and J. Arthur Seebach, Jr.
In contrast, general topologist
s, led by John L. Kelley
in 1955, usually did not assume T1, so they studied the separation axioms in the greatest generality from the beginning. Thus, they used the more complicated definitions for Ti, so that they would always have a nice property relating Ti to Tj. Then, for the simpler definitions, they used words (again, "regular" and "normal"). Both conventions could be said to follow the "original" meanings; the different meanings are the same for T1 spaces, which was the original context. But the result was that different authors used the various terms in precisely opposite ways. Adding to the confusion, some literature will observe a nice distinction between an axiom and the space that satisfies the axiom, so that a T3 space might need to satisfy the axioms T3 and T0 (e.g., in the "Encyclopedic Dictionary of Mathematics", 2nd ed.).
Since 1970, the general topologists' terms have been growing in popularity, including in other branches of mathematics, such as analysis. (Thus we use their terms in Wikipedia.) But usage is still not consistent.
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
, the separation axiom
Separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...
s have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept.
Origins
Before the current general definition of topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, there were many definitions offered, some of which assumed (what we now think of as) some separation axioms. For example, the definition given by Felix Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...
in 1914 is equivalent to the modern definition plus the Hausdorff separation axiom
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
The separation axioms, as a group, became important in the study of metrisability: the question of which topological spaces can be given the structure of a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
. Metric spaces satisfy all of the separation axioms; but in fact, studying spaces that satisfy only some axioms helps build up to the notion of full metrisability.
The separation axioms that were first studied together in this way were the axioms for accessible spaces, Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s, regular space
Regular space
In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
s, and normal space
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
s. Topologists assigned these classes of spaces the names T1, T2, T3, and T4. Later this system of numbering was extended to include T0, T2½, T3½ (or Tπ), and T5.
But this sequence had its problems. The idea was supposed to be that every Ti space is a special kind of Tj space if i > j. But this is not necessarily true. It depended on precisely how the definitions were phrased. For example, a regular space (called T3) does not have to be a Hausdorff space (called T2), at least not according to the simplest definition of regular spaces.
Different definitions
Every author agreed on T0, T1, and T2. For the other axioms, however, different authors could use significantly different definitions, depending on what they were working on. These differences could develop because, if one assumes that a topological space satisfies the T1 axiom, then the various definitions are (in most cases) equivalent. Thus, if one is going to make that assumption, then one would want to use the simplest definition. But if one did not make that assumption, then the simplest definition might not be the right one for the most useful concept; in any case, it would destroy the relation between Ti and Tj, allowing (for example) non-Hausdorff regular spaces.Topologists working on the metrisation problem generally did assume T1; after all, all metric spaces are T1. Thus, they used the simplest definitions for the Ti. Then, for those occasions when they did not assume T1, they used words ("regular" and "normal") for the more complicated definitions, in order to contrast them with the simpler ones. This approach reached its zenith in 1970 with the publication of Counterexamples in Topology
Counterexamples in Topology
Counterexamples in Topology is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.In the process of working on problems like the metrization problem, topologists have defined a wide variety of topological properties...
by Lynn A. Steen and J. Arthur Seebach, Jr.
J. Arthur Seebach, Jr.
J. Arthur Seebach, Jr was an American mathematician.He received his Ph.D. in 1968 from Northwestern University. He joined the faculty of Mathematics at St. Olaf College in 1965. He was Associate Editor of the American Mathematical Monthly from 1971 to 1986 and Editor of Mathematics Magazine from...
In contrast, general topologist
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
s, led by John L. Kelley
John L. Kelley
John Leroy Kelley was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis....
in 1955, usually did not assume T1, so they studied the separation axioms in the greatest generality from the beginning. Thus, they used the more complicated definitions for Ti, so that they would always have a nice property relating Ti to Tj. Then, for the simpler definitions, they used words (again, "regular" and "normal"). Both conventions could be said to follow the "original" meanings; the different meanings are the same for T1 spaces, which was the original context. But the result was that different authors used the various terms in precisely opposite ways. Adding to the confusion, some literature will observe a nice distinction between an axiom and the space that satisfies the axiom, so that a T3 space might need to satisfy the axioms T3 and T0 (e.g., in the "Encyclopedic Dictionary of Mathematics", 2nd ed.).
Since 1970, the general topologists' terms have been growing in popularity, including in other branches of mathematics, such as analysis. (Thus we use their terms in Wikipedia.) But usage is still not consistent.