TOPOLOGY ON R2
by
Emmanuel Kofi Kusi
MSC 2010: 54D30, 54D70
Keywords:Compactness, Base properties
Department Of Mathematics
University of Texas at El Paso.
April 6, 2018
Emmanuel Kofi Kusi TOPOLOGY ON R21 / 35
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Presentation on Topology, Slides of Mathematics
This is the presentation I did at the University of Texas at El Paso on Topological spaces on R^2 spaces. I did my undergrad at Kwame Nkrumah University of Science and Technology.
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TOPOLOGY ON R
2
by Emmanuel Kofi Kusi [email protected]
MSC 2010: 54D30, 54D Keywords:Compactness, Base properties
Department Of Mathematics University of Texas at El Paso.
April 6, 2018
Outline
(^1) Introduction (^2) Background of Study (^3) Problem Statement (^4) Objectives (^5) Definitions (^6) Applications (^7) Conclusion and Recommendation (^8) References
Background of Study
(^1) Topology, literally, means the study of position or location. Topology is the study of shapes including their properties, de- formations applied to them, mappings between them, and con- figurations composed on them. (^2) Topology is a very important area in mathematics. Basically, it is considered one of the three areas of pure mathematics (together with algebra and analysis). Recently topology has also become a very important part of applied mathematics with a lot of mathematician and scientists using concepts of topology to model and understand real world structures and situations. (^3) Topology is often described as rubber-sheet geometry. Tradi- tional geometry considered objects such as circles, triangles, planes and polyhedral as rigid, with distances between points and angles between faces or edges well defined. But in topol- ogy, these objects are treated as if they were made of rubber, capable of being deformed.
Background of Study cont’d
(^1) This makes distances between points and angles between edges irrelevant. The term topology was first used by Listing in his paper ”Vorstudien Topologie” (1847; ”Introductory Studies in Topology”) in 1847. (^2) The term was not commonly used and the discipline was not formerly defined until many years later. Poincare’s paper ”Anal- ysis Situs” in 1895, was the expression primarily used for this area of geometry. (^3) In the introduction to his paper ”Analysis Situs” Poincare wrote of the geometry-of-position philosophy. ”The proportions of the figures might be grossly altered, but their elements must not be interchanged and must conserve their relative situation. (^4) In other terms, one does not worry about quantitative prop- erties, but one must respect the qualitative properties, that is to say precisely those which are the concerns of Analysis Si- tus”(Sarkaria, 1999).
Objectives
Objectives (^1) To review some definitions and examples of Topology in R^2. (^2) To explore some applications of Topology in R^2.
Definitions
Topological Spaces A topological space is a set X together with τ which is a collection of subsets of X , called open sets satisfying the following axioms. (i) The ∅ and X are in τ. (ii) τ is closed under arbitrary union. (iii) τ is closed under finite intersection. The collection τ is called a topology on X. The elements of X are usually called points, though they can be any mathematical object.
Figure: A variety of Topological Spaces
Concept of Topology and Open sets
Example (3) On the real line, R, define a topology whose open sets the empty set and every set in R with a finite complement. See the figure below. For example, U = R−{ 0 , 3 , 7 } is an open set. We call this topology the finite complement topology on R and denote it by Rfc.
Figure: An open set in the finite complement on R
Definition Let X be a topological space and x ∈ X. An open set U containing x is called a neighborhood of x.
Concept of Topology and Open sets
Theorem (1) Let X be a topological space and let A be a subset of X. Then A is open in X if and only if for each x ∈ A, there is a neighborhood U of x such that x ∈ U ⊂ A. See figure below
Figure: The set A is open in X if and only if every point in A has a neighborhood U that lies in A
Closed Sets
Definition (^1) For each x in R^2 and > 0, define the closed ball of radius centered at x to be the set (^2) If [a, b] × [c, d] ⊂ R^2 is called a closed rectangle.
Theorem Closed balls and closed rectangles are closed sets in standard topology on R^2.
Closed Sets
Example(5) Consider R^2 with standard topology. In the figure below the top three sets are closed, and the bottom three are not.
Figure: In R^2 with the standard topology, the top three sets are closed, but the bottom three are not.
Interior and Closure of Sets
Definition Let A be a subset of a topological space X. The interior of A, denoted ◦A or Int(A), is the union of all open sets contained in A. The closure of A, denoted A¯ or Cl(A), is the intersection of all closed sets containing A.
Theorem Let X be a topological space, A be a subset of X and y be an element of X. Then y ∈ Int(A) if and only if if there exists an open set U such that y ∈ U ⊂ A y ∈ Cl(A) if and only if every open set containing y intersects A.
Example
In R^2 with the standard topology, let A be the lollipop that appears on the left in the figure below. Then Int(A) appears as in the middle of the figure, and Cl(A) appears as on the right.
Figure: The interior and closure of a set A in R^2
The Boundary of a subset
Example(6) Let A = [− 1 , 1] in the standard topology on R. As illustrated in the figure, we have Cl(A) = [− 1 , 1] and Int(A) = (− 1 , 1), and therefore ∂A = {− 1 , 1 }.
Figure: Determinining the boundary of [− 1 , 1] in R
The boundary of a Subset
Example(7) Let A be the subset of the plane appearing on the far left in the figure. Then, as pictured, Cl(A) is a closed ball, Int(A) is an open ball, and ∂A is a circle.
Figure: Determining the boundary of A in the plane.