Point-Set Topology: An Introduction

Branch of topology dealing with general topological spaces.

Point-set topology, also known as general topology, is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.

Open Sets, Closed Sets, and Limit Points

In topology, open sets are the fundamental building blocks of topological spaces. A set is open if, around each of its points, there is a neighborhood completely contained in the set. The complement of an open set is a closed set.

A limit point of a set is a point that can be "approached" by points of the set. More formally, a point x is a limit point of a set S if every neighborhood of x contains at least one point of S different from x itself.

Interior, Closure, and Boundary of a Set

The interior of a set consists of all points of the set that have a neighborhood completely contained in the set. The closure of a set is the smallest closed set containing the set, and it consists of all the points of the set and its limit points. The boundary of a set is the set of points that belong to the closure of the set but not to its interior.

Basis for a Topology

A basis (or base) for a topology on a set X is a collection of open subsets of X such that every open set can be written as a union of elements of the basis. The concept of a basis is fundamental in topology, as it allows us to describe and construct any topological space starting from a simpler collection of sets.

Subspace Topology

Given a topological space and a subset of it, we can define a topology on the subset, called the subspace topology, in which the open sets are the intersections of the open sets of the original space with the subset.

Continuous Functions and Homeomorphisms

A function between topological spaces is continuous if the preimage of every open set is open. This is the most basic and fundamental property that functions in topology should have. A homeomorphism is a continuous function that has a continuous inverse, i.e., it is a "topological isomorphism" between topological spaces.

Compactness and Connectedness in Point-Set Topology

Compactness and connectedness are two fundamental properties of topological spaces. A space is compact if every open cover has a finite subcover, and it is connected if it cannot be divided into two disjoint nonempty open sets. These properties have deep implications in the study of functions and in other areas of mathematics.

In conclusion, point-set topology provides the basic language and tools for the study of topology and its applications in various fields of mathematics. Understanding these concepts is essential for further study in topology, including the study of normed spaces.