TYPES OF TOPOLOGICAL SPACES AND THEIR INTERRELATIONSHIPS - Research Paper Example

? TYPES OF TOPOLOGICAL SPACES AND THEIR INTERRELATIONSHIPS TYPES OF TOPOLOGICAL SPACES AND THEIR INTERRELATIONSHIPS Introduction Mathematics of linear functions may not provide the necessary tools to handle different problems arising from nature. Indeed, the nature is chaotic and therefore the ‘good mathematics’ is not always sufficient. Topology is a branch of mathematics that deals with the study of spaces and shapes. Certainly, the human mind is suited for a two dimensional space. Consequently, it more difficult to study spaces of higher orders hence, the need to apply abstract tools. The beauty about mathematics is mathematicians avoid natural problems, instead they create and solve problems to represents the natural world. Therefore, much of the work done on topology is an artificial creation that resembles real world problem. Topology has significant applications in other branch of mathematics such as geometry and algebra. Major mathematical problems that can be solved using topology include continuity, connectedness, and boundary. The interesting aspect of topology is not the development of mathematical solutions, but how different mathematician approach a topology problem. This has led to the development of different topologies namely T1 – T4. This paper explores the different types of topology and their relationships. Definition 1.1. Let be a set and a collection of subsets of such that the following properties hold. I. The empty set and the space II. If , then III. If for , then The collection is referred to as a topology on and the pair is referred to as a topological space. Example 1.1. Let and, then is a topological space. The study of topological properties relies on neighborhoods or open sets that form any general space X. Thus, given an arbitrary set, there is more than one way of generating open sets/ neighborhoods that define its topology. Topologies can either be defined based on axioms of separation, compactness, or countability. This paper explores topologies resulting from the three topological definitions and the interrelationships between the topologies. 2. Spaces Based On Axioms of Separation Definition 2.1. In a Topological space, a sequence of elements of converges to if every open set containing contains a positive integer such that for . However, this definition does not endow a topological space with ‘nice’ properties similar to those found in metric spaces. For example in a metric space, every convergent sequence always converges to a unique limit. However, this is not necessarily true in topological spaces. To recover these properties, we need to supply enough open sets to the space. Thus, separation axioms classify topological spaces according to their sufficiency in open sets. Definition 2.2. A topological space is called a T0- space if for every two distinct points there exist an open set such that i. p lies in U and q does not lie in U. ii. q lies in U and p does not lie in U. Definition 2.3. T1 (Frechet) A topological space is called a T1- space if for every pair of points there exists such that Definition 2.4. T2 (Hausdorff) A topological space is called a T2-space or Hausdorff if for every pair of points there exists open sets such that, and. Definition 2.5. T3 (Regular) A T1 is called a T3 or a regular space if for every point and a closed set with there exists open sets such that and and. Definition 2.6. T3 1/2 (Completely Regular or Tychonoff) A T1 – space is called completely regular or Tychnoff if for every point and a closed set with there exist a continuous function such that and. Definition 2.7. T4 (Normal) A T1 – space is a T4 space if for every pair of disjoint closed sets A and B , there exists open sets such that , and . Remark 2.1 All T1 spaces are T0 but the converse is not true The discreet topology is T0 but not T1 All completely regular spaces are also T3 Every metric space is T4 Theorem 2.1. In any Hausdorff space, sequences have at most one limit It follows that every finite set is a T2 space is closed and therefore the topology induced by the Hausdorff property is stronger than the Cofinite Topology (where op-en sets are those with finite complements) 3. Spaces Based On Compactness The property of boundedness is essential in analysis. However, the property is not preserved under homeomorphism maps. Therefore, in order to generalize certain concepts of continuity notion of compactness is essential. Definition 3.1. A topological space is said to be compact if for every open covering of with there exist a finite sub-covering of such that. Remarks This definition has a major implication in . Under the usual topology in the usual topology, (Euclidian metric generated) closed set (closed interval) are compact. The definition of compactness as given above is applies for metric spaces; however, the definition can be extended to topological spaces through the concept of open sets. Definition 3.2. Para-Compactness: A topological space is said to be para-compact if for every open cover of, there exist a locally finite open covering say of such that (i.e. the collection has less open sets compared to). Theorem 3.1. Let be a continuous function such that, if is a compact space, then the image of in is also compact. Remark 3.1. This theorem implies that compactness is always preserved under continuous map. Thus, compactness is a topological invariant. The property of compactness sheds light on how various topological spaces relates. The following results highlight some of these relationships under the concept of compactness. Theorem 3.2. A compact subset of R with its usual metric is closed and bounded. Theorem 3.3. Any compact subset of a Hausdorff T2- space is closed. Remark: this theorem is equivalent to the statement that every subset of a T2- space is also a T2- space. Theorem 3.4. [The Heine–Borel Theorem.] Let R be given its usual topology (that is to say the topology derived from the usual Euclidean metric). Then the closed bounded interval [a, b] is compact. Theorem 3.5. Let [a, b] be given its usual topology (that is to say the topology derived from the usual Euclidean metric). Then the derived topology is compact. Theorem 3.6. Let and be topologies on the same space then, 1) If is compact, then is also compact 2) If and is Hausdorff, then so is. 3) If is compact and is Hausdorff, then 4. Topological spaces based on axioms of countability Definition 4.1. Let be a topological space, then is said to satisfy the first axiom of countability if every point of has a nbh basis consisting of at most countable many nbhs. The topological space is said to satisfy the second axiom of countability if there is an at most countable open basis for Remark 4.1 Under compactness, topological spaces are classified into compact and none compact spaces. The following collusions are true for topologies under the notion of compactness. 1) Any finite topological space is compact or any space that has a finite topology is compact 2) Any topological space that has the Cofinite Topology is compact 3) Any locally compact T2 space can be made compact 4) In the countable topology on , no finite set is compact Clearly, the first axiom of countability refers to a local point in while the second axiom of countability refers to the whole space. Consequently, a topological space that satisfies the second axiom of countability also satisfies the first axiom. Definition 4.2. Let be a topological space. If every open covering for of contains an utmost countable sub-covering for , then is called a Lindelof Space. Definition 4.3. Let be a topological space, if has a dense subset consisting of utmost countable points then is said to be a separable space. Theorem 4.1. Every subspace of a topological space satisfying the first or the second axiom of countability also satisfies the first or the second axiom of countability. Remark 4.2. A subspace of a first or second countable subspace is also countable The product of spaces satisfying the first or the second axiom of countability also satisfies the second or the first axioms of countability Theorem 4.2. If a topological space is second countable, then is first countable Lindelof and also separable Interrelationships between the Different Classes of Topologies Axioms of separation have strong concepts that distinguish between the different types of topologies. Under these axioms, it is possible to classify topologies as T2. In particular, the axioms attempt to answer the question of the degree of metrisability of a given topological space. For example, the Hausdorff space is separable. This implies that all distinct points in such a space can be separated using two disjoint open neighborhoods (elements of the topology). Thus, the axioms of separation show the richness of a topological space in terms of availability of open neighborhoods. On the other hand, compactness relates arbitrary spaces with finite sets. Under the consideration, a locally compact a locally compact Hausdorff space is T3. Moreover, a compact Hausdorff space is T4. The aspect of metrisability also links spaces developed under axioms of separation with those developed under axioms of countability. For example, If is a T3 space, which is second countable, and then it is metrisable (Urysohn’s Metrisation Theorem). REFERENCES 1. W. Stephen, General Topology, London, Dover Publications, 2004. 2. G. E. Bredon, Topology and Geometry (Graduate Texts in Mathematics), New York, Springer, 1997. 3. W. Fulton, Algebraic Topology, (Graduate Texts in Mathematics), New York, Springer, 1997. Read More