1.1 Definition and properties of topological spaces

Topological spaces are the foundation of topology, defining open sets and their properties. These concepts generalize familiar notions from real analysis, allowing us to study continuity and other properties in abstract spaces.

Bases simplify complex topologies, while subspace and product topologies extend these ideas. Separation axioms, like Hausdorff spaces, refine our understanding of how points in a space relate to each other, crucial for many advanced results.

Topological Spaces and Sets

Fundamental Concepts of Topological Spaces

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• Topological space consists of a set X and a collection of subsets T satisfying specific axioms
  • X and the empty set ∅ are elements of T
  • T is closed under arbitrary unions
  • T is closed under finite intersections
• Open set refers to any subset of X that belongs to the topology T
  • Forms the foundation for defining continuity in topological spaces
  • Generalizes the concept of open intervals in real numbers
• Closed set defined as a subset whose complement is open
  • Complements the notion of open sets in topological spaces
  • Includes the entire space X and the empty set ∅
• Neighborhood of a point x encompasses any open set containing x
  • Provides a way to describe local properties of points in the space
  • Can be open or closed, but must contain an open set containing x

Properties and Operations in Topological Spaces

• Interior of a set A comprises all points having A as a neighborhood
  • Denoted by Int(A) or A°
  • Always an open set and the largest open set contained in A
• Closure of a set A includes all points for which every neighborhood intersects A
  • Denoted by Cl(A) or Ā
  • Always a closed set and the smallest closed set containing A
• Boundary of a set A consists of points in the closure of A and its complement
  • Denoted by ∂A or Bd(A)
  • Can be expressed as ∂A = Cl(A) ∩ Cl(X\A)
  • Contains points that are "on the edge" of the set

Bases and Topologies

Constructing Topologies

• Basis for a topology generates the entire topology through unions of its elements
  • Simplifies the description of complex topologies
  • Must satisfy specific conditions (every point belongs to a basis element, intersection of basis elements is a union of basis elements)
• Subspace topology inherits its open sets from the topology of a larger space
  • Defined on a subset Y of a topological space X
  • Open sets in Y are intersections of open sets in X with Y
• Product topology extends the notion of topology to Cartesian products of spaces
  • Defined on X × Y where X and Y are topological spaces
  • Uses basis elements of the form U × V, where U and V are open in X and Y respectively

Advanced Topological Constructions

• Quotient topology arises from identifying points in a topological space
  • Defined on a set X/~ where ~ is an equivalence relation on X
  • A set U in X/~ is open if its preimage under the quotient map is open in X
  • Useful in constructing new spaces from existing ones (torus from a square)

Separation Axioms

Hausdorff Spaces and Separation Properties

• Hausdorff space satisfies a strong separation condition between points
  • For any two distinct points x and y, there exist disjoint open neighborhoods U of x and V of y
  • Ensures that distinct points can be separated by open sets
  • Generalizes the notion of "well-behaved" spaces in analysis and geometry
  • Many important topological properties hold only in Hausdorff spaces (compactness implies closed)