Closures - Spans - Generated Sets (𝐶𝑙(·), 𝑐𝑙(·), ·̅, ·‾)

Closure - Definitions

• Topological Spaces (Open Sets - Closed Sets - Clopen Sets - Closures)
• Totally Boundedness)

can be defined using any of the following equivalent definitions:

• 𝐶𝑙(𝑆) is the set of all points of closure of 𝑆
• 𝐶𝑙(𝑆) is the set 𝑆 together with all of its limit points:
  • each point of 𝑆 is a point of closure of 𝑆
  • each limit point of 𝑆 is a point of closure of 𝑆
• 𝐶𝑙(𝑆) is the intersection of all closed sets containing 𝑆
• 𝐶𝑙(𝑆) is the smallest closed set containing 𝑆
• 𝐶𝑙(𝑆) is the union of 𝑆 and its boundary 𝑏𝑑(𝑆)
• 𝐶𝑙(𝑆) is the set of all 𝑥∊𝑋 for which there exists a net (valued) in 𝑆 that converges to 𝑥 in (𝑋,𝜏)

Closure - Properties

• 𝐶𝑙(𝑆) is a closed superset of 𝑆
• the set 𝑆 is closed if and only if 𝑆 = 𝐶𝑙(𝑆)
• if 𝑆⊆𝑇 then 𝐶𝑙(𝑆)⊆𝐶𝑙(𝑇)
• if 𝐴 is a closed set, then 𝐴 contains 𝑆 if and only if 𝐴 contains 𝐶𝑙(𝑆)

Closure - Examples

In topological space (𝑋,𝜏)

• ⦰ = 𝐶𝑙(⦰)
• 𝑋 = 𝐶𝑙(𝑋)

Given ℝ and ℂ the standard topology (𝜏):

• If 𝑋=ℝ, then 𝐶𝑙_𝑋((0,1)) = [0,1]
• If 𝑋=ℝ, then 𝐶𝑙_𝑋(ℚ) = ℝ
  • the closure of the set ℚ of rational numbers is the whole space ℝ
  • we say that ℚ is dense in ℝ
• If 𝑋 is the complex plane ℂ = ℝ², then 𝐶𝑙_𝑋({𝑧∊ℂ : |𝑧| > 1}) = {𝑧∊ℂ : |𝑧| ≥ 1}
• If 𝑆 is a finite subset of a Euclidean space 𝑋, then 𝐶𝑙_𝑋(𝑆) = 𝑆

Closures depend on the underlying topology (𝜏)

• If 𝑋=ℝ is endowed with the standard topology, then 𝐶𝑙_𝑋((0,1)) = [0,1]
• If 𝑋=ℝ is endowed with the lower limit topology, then 𝐶𝑙_𝑋((0,1)) = [0,1)
• If 𝑋=ℝ is endowed with the discrete topology, then 𝐶𝑙_𝑋((0,1)) = (0,1)
  • in fact any subset 𝑆, 𝐶𝑙_𝑋(𝑆) = 𝑆
• If 𝑋=ℝ is endowed with the indiscrete topology, then 𝐶𝑙_𝑋((0,1)) = ℝ = 𝑋
  • in fact any subset 𝑆, 𝐶𝑙_𝑋(𝑆) = ℝ = 𝑋