Topology
- Preliminaries and Introduction
- Convergence in Topology
- Introduction to Normed Spaces
- Vector Spaces & Linear Operators
- Finite Dimensional Normed Spaces
- Infinite Dimensional Normed Spaces
- Compactness and Completeness in Normed Spaces
Preliminaries and Introduction
Notions in Set Theory
Branch of mathematics that studies sets, which are collections of objects.
Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It is a fundamental language of mathematics, and we use it as a foundational system for nearly every other branch of mathematics. In this unit, we will cover the basic operations and concepts in set theory.
Basic Set Operations
Union
The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. It is denoted by A ∪ B.
Intersection
The intersection of two sets A and B is the set of elements which are in both A and B. It is denoted by A ∩ B.
Difference
The difference of two sets A and B is the set of elements which are in A but not in B. It is denoted by A - B or A \ B.
Complement
The complement of a set A refers to elements not in A. If U is the universal set, then the complement of A is denoted by A' or A^c or U - A.
Cartesian Products
The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.
Power Sets
The power set of a set A is the set of all subsets of A, including the empty set and A itself. It is denoted by P(A).
Relations and Functions
A relation R from a set A to a set B is a subset of the Cartesian product A × B. If (a, b) is in R, we often write aRb.
A function from A to B is a special kind of relation for which each element of A is related to exactly one element of B.
Cardinality of Sets
The cardinality of a set A, denoted by |A|, is a measure of the "number of elements in the set". It could be finite or infinite.
Understanding these basic concepts in set theory is crucial for the study of topology. They provide the necessary tools to define and work with more complex structures in topology.