Infinite Collection of Sets

Topics: Calculus - Set - Topology

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(definition)

Let $A_1,A_2,\dots ,A_n,\dots$ be an infinite collection of sets. We define:

$${\cup }_{i=1}^{\infty }{A}_i=\{x\in X\mid x\in A_k,\text{ for some }k\}$$

$${\cap }_{i=1}^{\infty }{A}_i=\{x\in X\mid x\in A_k,\forall k\in \mathbb{N}\}$$

Example 1

Let:

• $A_1=(-1,1)$
• $A_2=(-\frac{1}{2},\frac{1}{2})$
• $A_3=(-\frac{1}{3},\frac{1}{3})$
• $A_4=(-\frac{1}{4},\frac{1}{4})$
• $A_5=(-\frac{1}{5},\frac{1}{5})$
• $⋮$
• $A_n=(-\frac{1}{n},\frac{1}{n})$
• $⋮$

We get that:

$${\cap }_{i=1}^{\infty }{A}_i=\{0\}$$

$${\cup }_{i=1}^{\infty }{A}_i=(-1,1)$$

Example 2

Let:

• $B_1=(0,1)$
• $B_2=(1,2)$
• $B_3=(2,3)$
• $⋮$
• $B_n=(n-1,n)$
• $⋮$

We get that:

$${\cap }_{i=1}^{\infty }{B}_i=\varnothing$$

$${\cup }_{i=1}^{\infty }{B}_i=(0,\infty )-\mathbb{N}$$

Union of Open Sets is Open §

(theorem)

Let $A_1,A_2,\dots ,A_i,\cdots \subseteq {\mathbb{R}}^n$ be an infinite collection of open sets. Then, we have that ${\cup }_{i=1}^{\infty }{A}_i$ (their union) is also an open set.

Intersection of Closed Sets is Closed §

(theorem)

Let $A_1,A_2,\dots ,A_i,\cdots \subseteq {\mathbb{R}}^n$ be an infinite collection of closed sets. Then, we have that ${\cap }_{i=1}^{\infty }{A}_i$ (their intersection) is also a closed set.