Closure is Monotonic

Last updated Nov 6, 2022

Let $X$ be a Topological Space. Suppose $A \subset B \subset X$. Then $\text{cl} A \subset \text{cl} B$.

Limit Points of a subset are Limit Points of the original Set so Limit Points of $A$ are Limit Points of $B$. Since Closure of a Set is all its Limit Points, $$\text{cl} A = {x \in X : x \text{ is a limit point of } A} \subset {x \in X : x \text{ is a limit point of } B} = \text{cl} B.$$ $\blacksquare$

1. If $B$ is Closed, then $\text{cl} A \subset \text{cl} B = B$ so $\text{cl} A \subset B$.