Point is Isolated iff belongs to Set less Derivative
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Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $H \subseteq S$.
Let $x \in S$.
Then:
- $x$ is an isolated point in $H$
- $x \in H \setminus H'$
where
- $H'$ denotes the derivative of $H$.
Proof
$x$ is an isolated point in $H$
$\iff$ $x \in H$ and $x$ is not an accumulation point of $H$ by Point is Isolated iff not Accumulation Point
$\iff$ $x \in H$ and $x \notin H'$ by definition of derivative
$\iff$ $x \in H \setminus H'$ by definition of set difference.
$\blacksquare$
Sources
- Mizar article TOPGEN_1:20