Topological Spaces (Neighborhoods of a Point/Set)

Neighborhoods 𝑁_𝜏(·) ^{(general definition)}

• given a topological space (𝑋,𝜏) where:
  • 𝑋 is a set
  • 𝜏 is a topology on 𝑋
• a neighborhood of point 𝑎∊𝑋, denoted as 𝑁_𝜏(𝑎), is a subset 𝑆 of 𝑋 that contains an open set 𝑂 (i.e. 𝑂∊𝜏) where 𝑂 contains 𝑎:

  • 𝑎∊𝑂⊆𝑆⊆𝑋 → 𝑁_𝜏(𝑎)=𝑆

• a neighborhood of set 𝐴⊆𝑋, denoted as 𝑁_𝜏(𝐴), is a subset 𝑆 of 𝑋 that contains an open set 𝑂 (i.e. 𝑂∊𝜏) where 𝑂 contains 𝐴:
  
  • 𝐴⊆𝑂⊆𝑆⊆𝑋 → 𝑁_𝜏(𝐴)=𝑆