Definition
Consider a subset of a Topological Space . The closure of is defined as
\bar{A} = \operatorname{cl}A :=& \{x \in X | \forall \mathcal{N}_{x},\ \mathcal{N}_{x} \cap A \neq \emptyset\}\\ =& \bigcap \{ C | (X \setminus C) \in \mathcal{T}\ \text{and}\ A \subset C\}\\ =& A \cup A'\\ =& A \cup \partial A \end{aligned}$$ where $\mathcal{N}_{x}$ is a [[Neighborhood]] of $x$. The closure of the $A$ can be defined in any of the following equivalent ways: - $\operatorname{cl}A$ is the set of all points of closure of $A$. - $\operatorname{cl}A$ is the intersection of all [[Closed Set|closed sets]] containing $A$. - $\operatorname{cl}A$ is the union of $A$ and the set of all its [[Limit Point|limit points]] $A'$. - $\operatorname{cl}A$ is the union of $A$ and its [[Boundary]] $\partial A$ # Facts > ![[Pasted image 20241115121613.png|300]] > > Consider a subspace topology $(Y, \mathcal{T}_{Y})$ of a [[Topological Space]] $(X, \mathcal{T})$ and a subset $A \subset Y$. Then, > $$\operatorname{cl}_{Y} A = \operatorname{cl}_{X} A \cap Y$$ > For subsets $A, B$ of a [[Topological Space]] $(X, \mathcal{T})$, > $$A \subset B \Longrightarrow \bar{A} \subset \bar{B}$$ > For subsets $A, B$ of a [[Topological Space]] $(X, \mathcal{T})$, > $$\overline{A \cup B} = \bar{A} \cup \bar{B}$$