Homework 1 (first part) [adapted from latex HW of Colin McKinney] 1.0 Show X − ⋂_{i=1}^{n} U_{i} = ⋃_{i=1}^{n}(X − U_{i}) x ∈ X − ⋂_{i=1}^{n} U_{i} iff x ∈ X and x ∉ ⋂_{i=1}^{n} U_{i} iff x ∈ X and x ∉ U_{i} for some i iff x ∈ X − U_{i} for some i iff x ∈ ⋃_{i=1}^{n}(X − U_{i}). 1.1 Let X be a toplogical space; let A be a subset of X. Suppose that for each x ∈ A there is an open set U_{x} containing x such that U_{x} ⊂ A. Show that A is open in X. Proof: For each x ∈ A, take an open set U_{x} containing x such that U_{x} ⊂ A (U_{x} exists by hypothesis). Hence ∪_{x∈A}U_{x} is also open (by defn of topology). ∪_{x∈A}U_{x} = A, and so A must also be open.