Set of Condensation Points is Subset of Derivative
Jump to navigation
Jump to search
Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A$ be a subset of $S$.
Then:
- $A^0 \subseteq A'$
where
- $A^0$ denotes the set of condensation points of $A$
- $A'$ denotes the derivative of $A$
Proof
Let $x \in A^0$.
By definition of set of condensation points:
- $x$ is condensation point of $A$
By definition of condensation point:
- $x$ is limit point of $A$
By definition of derived set:
- $x \in A'$
$\blacksquare$
Sources
- Mizar article TOPGEN_4:50