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CONVEX TOPOLOGY
Invented by Douglas Moreman

Basic Concepts
Perhaps you are not a topologist and would appreciate even the briefest of examples.
The word "domain" will be a tool for getting at the essence of the meaning of "limit" as used in, for example, Calculus.
Suppose the context of our example(s) is Space which is either 1-Space (the set of all real numbers) or 2-Space (the Euclidian Plane).
In 1-Space, the set of all points (i.e. numbers) between two numbers is a "region" and in the context of 2-Space, the word "region" means the interior of a circle. In these contexts, the word "domain" can be taken to mean "a union of regions." The collection of all domains is a "topology" for the space. The concept of "limit," fundamental to Calculus, can be defined via the word "domain."
A "topology" (in the sense used on this web page) is said to be determined by meanings of the words "point" and "domain" such that
* every domain is a point-set of Space,
* every region is a domain,
* every set that is the union of domains is a domain,
* every set that is the intersection of finitely many domains is a domain, and
* if D is a domain and P is a point of D then there exist a region R such that R contains P and lies in D.
Definition. The statement that P is a limit point of the point-set M means that every domain that contains P contains a point of M distinct from P.
In 1-Space (where "points" are numbers) the number 0 is the only limit-point of the set M consisting of every number that is the reciprocal of a positive integer. (this relates to the fact that the sequence 1/1, 1/2, 1/3, ... converges to 0)
Note, as an example, that while 0 is a limit-point of M, 0 is not a member of the set M. M is not "closed" - there exists a limit point of the set M that is not a member of M.
Definition. A point-set M is said to be closed provided there does not exist a limit-point of M that is not a member of M.

Definition. If M is a point set then by the closure of M, denoted by ~M, is meant the set consisting of every point of M and every limit point of M.

In the context of a topology, one can prove that if M is a point set then ~M is closed.

The idea of "convexity" can be used to discriminate between the interior of a circle and the interior of a 2-Space version of a kidney-bean.
Suppose "between" has its customary sense.
Definition. The statement that a point set M is convex means that if A and B are two points of M then every point between A and B is also a point of M; or, in other words (this is an English-lesson) there does not exist a point P such that P is between two points of M but P is not a member of M.
If, in 1-Space, M is the set whose only member is 0 then M is a point-set and M is both closed and convex.

Convex Topology may be thought of as an abstraction of, a generalization of, the two notions of a point-set being closed and a point set being convex.

CONVERGENCE AND COMPACTNESS
Suppose 1) the word point has a meaning and S is the set of all points,
2) the word domain has such a meaning that
a) S is a domain,
b) each set that is the common part of two domains is a domain,
c) each set that is a union of domains is a domain, and
d) if P and Q are two points then there is a domain that contains P but does not contain Q,
3) the word convex has such a meaning that the collection C of all convex point sets is a convexity for S in the sense that
a) S is convex, and
b) each set that is a common part of convex point sets is itself convex, and
4) B is a collection of domains such that every domain is a union of elements of B. (Perhaps B is the set of all domains.)

If each of H and K is a point set then the common part of H and K will be denoted by H.K.

If M is a point set then by the convex hull of M (denoted by C(M)) is meant the common part of all convex point sets that contain M.

Each point set M has a convex hull and the convex hull of M is convex.
If G is a topology for S then by a basis for G is meant a sub-collection H of G such that if P is a point of an element D of G then some element of H contains P and is a subset of D. Such terms as "limit point" and "closed" will be intended, unless otherwise specified, in reference to the topology for which B is a basis. If M is a point set then the closure of M is denoted by ~M. If G is a collection of point sets then the common part of all topologies for S that have G as a subcollection is itself a topology for S.

If G is a collection of point sets then by the minimum topology for S that has G as a subcollection is meant the common part of all topologies for S that have G as a subcollection.

Definition. By the convex topology of S,B,C is meant the minimum topology for S that contains each set that is the complement of the closure of a convex point set.

Note. If S is a normed linear space, B is the norm topology of S, and C is the collection of all sets that are convex relative to the linear structure of S then the convex topology of S is the so-called weak topology of S.

Definitions. The statement that R is an Omega1 region means that R is S or R is the complement of the closure of a convex set. The statement that R is an Omega region means that R is an Omega1 region or the common part of finitely many Omega1 regions. The statement that R is an Omega domain means that R is a union of Omega regions. By "Omega" is meant the collection of all Omega domains.

Theorem C1.1. R is an Omega region if and only if R is S or R is the complement of the sum of the closures of finitely many convex sets.
Proof. Part 1) Suppose R is an Omega region but R is not S. There exists a finite sequence R1, R2, ..., R(t) each term of which is an Omega1 region and such that R is the common part of the terms of that sequence.
There exists a sequence K(1),K(2),...,K(t) of convex point sets such that
R(1)= S- ~K(1), R(2) = S- ~K(2),..., R(t) = S- ~K(t).
R = (S-~K(1)) . (S-~K(2)) .... .(S-~K(t)), which is
S - (~K(1)+ ~K(2)+ ... + ~K(t)); so that, R is the complement of the sum of the closures of finitely many convex sets.

Part 2) S is an Omega region. Now, suppose R is the complement of the sum of the closures of finitely many convex sets K(1),K(2),... , K(t).
S-(K(1)+K(2)+.. .+K(t)) is (S-K(1)).(S-K(2)).....(S-K(t)) so that R is the common part of finitely many Omega1 regions.

Theorem C1.2. If R is the common part of two Omega regions then R is an Omega region.

Proof. Suppose R is the common part of two Omega regions R(1) and R(2). There exist finite collections H and K of closures of convex sets such that R(1) = S-H* and R(2) = S-K*. R(1)*R(2) = S-(H*+K*); so that ,
R = S-(H+K)*. R is the complement of the sum of the closures of finitely many convex sets. R is, therefore , an Omega region.

Theorem C1.3. Omega is the convex topology of S,B,C and the collection of all Omega regions is a basis for Omega.
Proof. Part 1) Omega is a topology for S. For:
a) S is an element of Omega.
b) If G is a subcollection of Omega then G* is an element of Omega. For, suppose G is a subcollection of Omega. Each element g of G is the sum of the elements of a collection Hg of Omega regions. Let H denote the union of all such collections Hg. If P is a point of H* then, for some element g of G, P belongs to Hg so that P belongs to g and therefore P belongs to G*. If P is a point of G* then P belongs to an element g of G, P belongs to an element of Hg, and P belongs to H*.
G* is H*. Therefore,
G* is an Omega domain, and G* is an element of Omega. c) If D and R are intersecting elements of Omega then the intersection, D*R, is an element of Omega. For, there exist collections H and K of Omega regions such that D is H* and R is K*.
If P is a point of D*R then P belongs to an element of the collection W consisting of every point set that is the common part of an element of H and an element of K. If P belongs to an element of W then P belongs to D*R.
W* is D*R. It is a consequence of Theorem CI.2 that each element of the collection W is an Omega, region. D*R is the sum of Omega regions and is, therefore, an element of Omega.

Part 2) The collection of all Omega regions is a basis for Omega. For, suppose P is a point of an Omega domain D. Since D is a union of Omega regions, there is an Omega region that contains P and lies in D.

Part 3) If G is a topology for S such that every Omegal region is an element of G then Omega is a subcollection of G. If every Omega region is an element of the topology G then every Omega domain is an element of G. So, suppose that R is an region. There exists a finite collection H such that each element of H is an Omega1 region and R is the point set to which P belongs if and only if P belongs to every set in H. H is a finite subcollection of G. Since G is a topology, R is an element of G.

Theorem CI.4. The convex topology of S,B,C is both the minimum topology for S that contains each set that is the complement of the closure of an element of C and the minimum topology for S under which the closure relative to B of convex point sets are closed.