Interior, Boundary, and Exterior

 

Def 1

Let $(X, \mathscr{T})$ be a topological space  and  $A \subseteq X$.

1.  $x$ is called interior point of $A$

$\Leftrightarrow$  $\exists U \in \mathscr{T}$  s.t.  $x \in U \subseteq A$

 

2.  The set of all interior point of $A$ is called interior of $A$.

We write $A^{\circ}$  or  $\mathrm{Int}(A)$

 

3.  $x \in X$ is called boundary point of $A$

$\Leftrightarrow$  $x \in \overline{A} \cap \overline{(X \setminus A)}$

 

4.  The set of all boundary point of $A$ is called boundary of $A$ (or frontier of $A$).

We write $\mathrm{Fr}(A)$

 

5.  $\mathrm{Int}(X \setminus A)$ is called exterior of $A$.

We write $\mathrm{Ext}(A)$

 

 

 

Thm 1

Let $(X, \mathscr{T})$ be a topological space  and  $A \subseteq X$.

$\displaystyle \bigcup_{\begin{gather*} U \subseteq A \\ U :  \text{open set} \end{gather*}} U = \mathrm{Int}(A)$

 

  $x \in \displaystyle \bigcup_{\begin{gather*} U \subseteq A \\ U : \text{open set} \end{gather*}} U$     $\Leftrightarrow$     $\exists U \in \mathscr{T}$  s.t.  $x \in U \subseteq A$     $\Leftrightarrow$     $x \in \mathrm{Int}(A)$

 

 

 

Thm 2

Let $(X, \mathscr{T})$ be a topological space  and  $A \subseteq X$.

$A$ is an open set  $\Leftrightarrow$  $\mathrm{Int}(A) = A$

 

pf)

$\Rightarrow)$

  Since $A$ is open set,

$A \subseteq \displaystyle \bigcup_{\begin{gather*} U \subseteq A \\ U : \text{open set} \end{gather*}} U = \mathrm{Int}(A) \subseteq A$

 

$\Leftarrow)$

$A = \mathrm{Int}(A) = \displaystyle \bigcup_{\begin{gather*} U \subseteq A \\ U : \text{open set} \end{gather*}} U \in \mathscr{T}$

 

 

 

Thm 3

Let $(X, \mathscr{T})$ be a topological space  and  $A \subseteq X$.

$x \in \mathrm{Fr}(A)$  $\Leftrightarrow$  $\forall U \in \mathscr{T}$ containing $x$,  $U \cap A \neq \varnothing$  and  $U \cap (X \setminus A) \neq \varnothing$

 

  $x \in \mathrm{Fr}(A)$  $\Leftrightarrow$  $x \in \overline{A} \cap \overline{(X \setminus A)}$

                          $\Leftrightarrow$  $x \in \overline{A}$  and  $x \in \overline{(X \setminus A)}$

                          $\Leftrightarrow$  $\forall U \in \mathscr{T}$ containing $x$,  $U \cap A \neq \varnothing$  and  $U \cap (X \setminus A) \neq \varnothing$

 

 

 

Thm 4

$X$ : topological space, $A \subseteq X$

$X = \mathrm{Int}(A) \sqcup \mathrm{Ext}(A) \sqcup \mathrm{Fr}(A)$

($\sqcup$ is disjoint union.)

 

  Show 1 : $\mathrm{Int}(A) \cap \mathrm{Ext}(A) = \varnothing$

  Since $\mathrm{Int}(A) \subseteq A$  and  $\mathrm{Ext}(A) \subseteq X \setminus A$,

  $\mathrm{Int}(A) \cap \mathrm{Ext}(A) = \varnothing$

 

  end

 

 

  Show 2 : $x \notin \mathrm{Int}(A)$  and  $x \notin \mathrm{Ext}(A)$  $\Leftrightarrow$  $x \in \mathrm{Fr}(A)$

  $x \notin \mathrm{Int}(A)$  and  $x \notin \mathrm{Ext}(A)$ 

       $\Leftrightarrow$  $\forall U \in \mathscr{T}$ containing $x$,  $U \nsubseteq A$  and  $U \nsubseteq X \setminus A$

       $\Leftrightarrow$  $\forall U \in \mathscr{T}$ containing $x$,  $U \cap (X \setminus A) \neq \varnothing$  and  $U \cap A \neq \varnothing$

       $\Leftrightarrow$  $x \in \mathrm{Fr}(A)$

 

  end