Set of Condensation Points is Monotone
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B$ be subsets of $S$.
Then:
- $A \subseteq B \implies {A^0} \subseteq B^0$
where
- $A^0$ denotes the set of condensation points of $A$
Proof
Assume
- $A \subseteq B$
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 $A$ such that $\forall U \in \tau: A \cap U$ is uncountable
Thus by Limit Point of Subset is Limit Point of Set:
- $x$ is limit point of $B$
Let $U \in \tau$.
By definition of condensation point:
- $A \cap U$ is uncountable
By Set Intersection Preserves Subsets/Corollary:
- $A \cap U \subseteq B \cap U$
Thus by Subset of Countable Set is Countable
- $B \cap U$ is uncountable
Then by definition:
- $x$ is condensation point of $B$
Thus by definition of set of condensation points:
- $x \in B^0$
$\blacksquare$
Sources
- Mizar article TOPGEN_4:52