Definition

Consider a Topological Space $(X, T)$ . A set $C$ is closed if $X ∖ C \in T$

The complement of an Open Set is called a closed set.

Facts

$\emptyset$ and $X$ are closed

$\forall closed {C_{i}}_{i \in I}, i \in I ⋂ C_{i} is closed$

$\forall closed {C_{i}}_{i = 1}^{n}, i = 1 ⋃ n C_{i} is closed$

Consider a Subspace Topology $(Y, T_{Y})$ of a Topological Space $(X, T)$ and a subset $A \subset Y$ . Then, $(Y ∖ A) \in T_{Y} ⟺ (\exists (X ∖ C) \in T s.t. A = C \cap Y)$

Consider a Subspace Topology $(Y, T_{Y})$ of a Topological Space $(X, T)$ and a subset $A \subset Y$ . Then, $((Y ∖ A) \in T_{Y} and (X ∖ Y) \in T) ⟹ X ∖ A \in T$

Consider a Topological Space $(X, T)$ and a subset $A \subset X$ . Then, $(X ∖ A) \in T ⟺ A^{'} \subset A$ where $A^{'}$ is the set of all limit points of $A$

Consider a Topological Space $(X, T)$ and a subset $A \subset X$ . If the subset is equal to its Closure, it is closed. $(X ∖ A) \in T ⟺ \overset{ˉ}{A} = A$

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