1.2 Open and closed sets

Open and closed sets form the foundation of topological spaces, defining the structure of these abstract mathematical objects. These sets help us understand how points relate to each other and provide a framework for analyzing continuity, compactness, and other key concepts in topology.

Understanding open and closed sets is crucial for grasping the nature of topological spaces. They allow us to generalize ideas from metric spaces to more abstract settings, providing tools to study properties that remain unchanged under continuous transformations.

Open vs Closed Sets

Defining Topological Spaces and Sets

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• Topological space consists of a set X and collection of subsets τ satisfying specific axioms
• Open set belongs to topology τ
• Closed set defined as complement of an open set
• Interior of set A represents largest open set contained within A
• Closure of set A represents smallest closed set containing A
• Boundary of set A includes points in closure of A but not in interior
• Sets can be open, closed, both (clopen), or neither
• Metric spaces characterize open sets using open balls
• Closed sets in metric spaces defined using sequences and limit points

Examples in Topological Spaces

• Real number line with standard topology
  • Open sets include intervals (a, b) where a < b
  • Closed sets include intervals [a, b] where a ≤ b
• Discrete topology on set X
  • Every subset of X is both open and closed (clopen)
• Indiscrete topology on set X
  • Only ∅ and X are both open and closed
• Finite complement topology on infinite set X
  • Open sets include X and all subsets with finite complements
  • Closed sets include ∅ and all finite subsets of X

Properties of Open and Closed Sets

Fundamental Set Properties

• Empty set ∅ and entire space X both open and closed in any topological space
• Arbitrary union of open sets yields open set
• Finite intersection of open sets produces open set
• Arbitrary intersection of closed sets results in closed set
• Finite union of closed sets creates closed set
• De Morgan's laws relate properties of open and closed sets
  • Complement of union equals intersection of complements
• Proofs utilize definitions of open and closed sets and set theory operations

Examples of Set Operations

• Union of open intervals on real line: $(0, 1) ∪ (2, 3) = (0, 1) ∪ (2, 3)$ (open)
• Intersection of open balls in metric space: $B(x, r) ∩ B(y, s)$ (open)
• Union of closed intervals: $[0, 1] ∪ [2, 3] = [0, 1] ∪ [2, 3]$ (closed)
• Intersection of closed sets in $ℝ^2$: $\{(x, y) | x^2 + y^2 ≤ 1\} ∩ \{(x, y) | y ≥ x\}$ (closed)

Open and Closed Sets: Relationships

Complementary Relationships

• Complement of open set closed, complement of closed set open
• Closure of set A equals intersection of all closed sets containing A
• Interior of set A equals union of all open sets contained in A
• Set open if and only if equal to its interior
• Set closed if and only if equal to its closure
• Boundary of set A closed, expressed as intersection of closure of A and closure of its complement
• Every open set expressed as complement of closed set, and vice versa
• Open and closed sets exhibit duality in many theorems

Examples of Set Relationships

• Real line topology
  • Interior of [0, 1] equals (0, 1)
  • Closure of (0, 1) equals [0, 1]
  • Boundary of (0, 1] equals {0, 1}
• Discrete topology on set X
  • Interior of any subset A equals A
  • Closure of any subset A equals A
  • Boundary of any subset A equals ∅
• Finite complement topology on infinite set X
  • Interior of finite set A equals ∅
  • Closure of cofinite set B equals X
  • Boundary of finite set C equals C

Applying Open and Closed Sets

Problem-Solving Techniques

• Use open set definition to prove given set in topological space open or not open
• Apply closed set properties to determine if set closed in specific topological space
• Construct proofs using relationships between open sets, closed sets, and complements
• Utilize interior, closure, and boundary concepts to analyze set properties in topological spaces
• Apply open and closed set properties to prove complex topological concepts (continuity, compactness)
• Solve problems involving intersection and union of open and closed sets
• Use counterexamples to disprove false statements about open and closed sets

Application Examples

• Prove continuity of function $f: ℝ → ℝ$ by showing preimage of open set open
• Show compactness of closed interval [a, b] in ℝ using properties of closed sets
• Determine whether set $A = \{(x, y) | x^2 + y^2 < 1\}$ open or closed in $ℝ^2$
• Prove connectedness of interval (a, b) using properties of open and closed sets
• Construct counterexample to statement "union of closed sets always closed" using infinite collection of closed intervals