Derived Set Preserves Set Inclusion

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $B \subseteq A \subseteq S$.

Then:

$B' \subseteq A'$

where $A'$ and $B'$ are the derived sets in $T$ of $A$ and $B$ respectively.

Proof

Let:

$x \in B'$

By the definition of derived set:

$x$ is a limit point of $B$.

Note that $B \subseteq A$.

From Limit Point of Subset is Limit Point of Set:

$x$ is a limit point of $A$.

So, $x \in A'$.

That is:

$B' \subseteq A'$

$\blacksquare$