MTH634 Midterm Subjective and Objective by VUBuddy

DATA for MID TERM ---- MTH634 ---- Module# 001 to 085

Topology is combination of two words “Topes (Greeks) means Place” and “Logy (Latin) means Study/Analysis”. Topology is a kind of Math in which we study shapes. In Geometry we care about exact measurements such as lengths, angles and areas. In Topology we do not care about measurements. Geometrically rectangle, triangle, circle and square are different shapes. By topologically rectangle, triangle, circle and square are same shapes. In topology we care about arrangement of shapes, relative positions of shapes and Functions/Deformations between the shapes. By definition topology means “Geometry of positions”. In topology two shapes are considered same if one shape can be deformed to other shape. For example a circle can be deformed into a square so topologically circle and square are same shapes. A line and a circle are different shapes. In fact Topology is a matter of nature of study. Topological study is more qualitative in nature rather than quantitative.

Let X be a set and Power set of X is denoted by P(X). Power set is the collection of all sub sets of a given set. Usually we use the word class for a set of sets. ∅ and set itself are also members of power set. If X is a finite set with “n” elements then number of sets in powers set = 2n Let X be a non-empty set and 𝝉 be a subclass of P(X) then 𝝉 is called a topology on X if 𝝉 satisfied the

Topology on a set can also be defined by starting only 2nd and 3rd axioms.

More than one topology can be defined on a set. Not every collection of subsets of a set is eligible to be a topology on the set unless that collection does not satisfy all the three stated conditions of given definition.

For any set there are always two special topologies. (i) Indiscrete Topology (ii) Discrete Topology Indiscrete Topology: A topology on a set X consists of empty set and the set itself is called Indiscrete Topology. It is also called Trivial Topology. Indiscrete topology is the smallest topology that can be defined on some set X. Discrete Topology: A topology on a set X consists of all subsets of X is called Discrete Topology i.e. 𝝉= 𝑷(𝑿) Discrete topology is the largest topology that can be defined on some set X. Cofinite Topology: A topology on a set X is the collection of all subsets U of X such that Uc is either finite or Cofinite topology is also known as “Finite Complement Topology”.

Discrete Space: A set X with discrete topology is called “Discrete Topological Space”. Indiscrete Space: A set X with indiscrete topology is called “Indiscrete Topological Space”.

Arbitrary intersection of open sets may or may not be open.

Closed Set: Let (X,T) be a topological space. A subset “A” of “X” is closed iff the complement of “A” in “X” is

Let (X,T) be a topological space. Consider the collection A={A ⊂ X | Ac ∈𝝉 } i.e. “A” is the set of all those subsets of X whose complement is open. Simply: “Collection of all closed subsets”.

all closed subsets of X. Intersection of topologies on a set X is always a topology on set X. Union of topologies on a set X may or may not be a topology on X. The real number set ℝ is one of the most studies or dominant object in Mathematics. Usual Topology on ℝ: The set of all open sets of ℝ is Tu called Usual Topology on ℝ. Real Plan ℝ𝟐: The real plan is the Cartesian product of two real lines i.e. ℝ× ℝ. The real plane is written as ℝ2

distance from (x,y) is less than “r” where “r” is positive real number. Let T1 and T2 be two topologies on a non empty set X then T1 is finer than T2 iff 𝑇1 ⊆𝑇2 Indiscrete topology is the weakest topology on a set X. Discrete topology is the finest topology on a set X. Two topologies on a set X are comparable iff one is finer (weaker) than the other. Not all topologies on a set X are comparable. Limit Point: Let (X,T) be a topological space and “A” be a subset of X. A point 𝑥∈𝑋 is a limit point of A iff every open set Ux containing x contains a point of A different from x. Limit Point is also called accumulation point or derived point. [ALSO PREPARE PAGE#50-52]

Derived Set: Let (X,T) be a topological space and A be a subset of X then the derived set of A is denoted by 𝑨

Derived set of A may or may not be subset of A. Derived set of empty set is empty. In a discrete space X the derived set 𝑨 of any subset A of X is empty.

Prepare Page # 61-62 A subset “A” of a topological space X is closed iff 𝐴 ⊂𝐴 Let (X,T) be a topological space and “A” be a subset of X. The closure of “A” is denoted by 𝐴 and it is the

X containing A.

Let A be a subset of a topological space X then = 𝐴 Prepare Page#78-79

Let A be a subset of a topological space X then A is said dense in X iff 𝐴 =X

Q is dense in ℝ. N is not dense in ℝ. X is dense in itself. A proper closed subset of X can never dense in X.

Boundary points are also called frontier Points. bd(A) is closed. Boundary of a set A is empty iff A is both open and close. Let P be a point in a topological space (X,T). A subset N of X is a neighborhood of p iff there exists an open set

Let P be a point in a topological space (X,T then N is neighborhood of “p” iff “p” is an interior point of N. A neighborhood N of a point “p” is called an open neighborhood iff N is a neighborhood of all its points i.e. N is an open set. The class of all neighborhoods of 𝑝∈𝑋 is called neighborhood system of “p”. It is denoted by “Np”. Let Np is the neighborhood system of “p” then Np is non-empty. Let Np is the neighborhood system of “p” then Np is closed under finite set intersection. Let Np is the neighborhood system of “p” then every superset of a member of Np belongs to Np.

Subspace Topology: Let A be a nonempty subset of a topological space (X,T) Then the collection TA={V=U⋂𝐴|𝑈∈𝑇} of subsets of defines a topology on A. TA is called subspace topology or relative topology on A relatives to (X,T).