Topology/Points in Sets

Some Important Constructions

Let $A$ be an arbitrary subset of $X$.

Closure

• A point x is called a point of closure of a set A if for every neighbourhood $U$ of x, $U\cap A\ne \mathrm{\varnothing }$
• Define the closure of $A$ to be the intersection of all closed sets containing $A$, denoted $\mathrm{C}\mathrm{l}(A)$ (some authors use $\overline{A}$). The closure has the nice property of being the smallest closed set containing $A$. Each neighborhood (nbd) of every point in the closure intersects $A$.

Interior

• We say that $x$ is an internal point of $A$ iff There is an open set $U$, $x\in U$ and $U\subseteq A$
• Define the interior of $A$ to be the union of all open sets contained inside $A$, denoted $\mathrm{I}\mathrm{n}\mathrm{t}(A)$ (some authors use $A^{\circ }$). The interior has the nice property of being the largest open set contained inside $A$. Every point in the interior has a nbd contained inside $A$.

Note that a set $A$ is Open iff $A=Int(A)$

Exterior

• Define the exterior of $A$ to be the union of all open sets contained inside the complement of $A$, denoted $\mathrm{I}\mathrm{n}\mathrm{t}(X\setminus A)$. It is the largest open set inside $X\setminus A$. Every point in the exterior has a nbd contained inside $X\setminus A$.

Boundary

• Define the boundary of $A$ to be $\mathrm{C}\mathrm{l}(A)\setminus \mathrm{I}\mathrm{n}\mathrm{t}(A)$, denoted $\mathrm{B}\mathrm{d}(A)$ (some authors prefer $\partial A$). The boundary is also called the frontier. It is always closed since it is the intersection of the closed set $\mathrm{C}\mathrm{l}(A)$ and the closed set $X\setminus \mathrm{I}\mathrm{n}\mathrm{t}(A)$. It can be proved that $A$ is closed if it contains all its boundary, and is open if it contains none of its boundary. Every nbd of every point in the boundary intersects both $A$ and $X\setminus A$. All boundary points of a set A are obviously points of contact of a set A.

Limit Points

• A point x is called a limit point of a set A if for every neighborhood $U$ of x, $(U\setminus \{x\})\cap A\ne \mathrm{\varnothing }$. All limit points of a set A are obviously points of closure of the set A.

Some Basic Theorems

The A⊆Cl(A) since if a point x is within A, then for all neighborhoods N of that point, N∩A≠Ø because it contains x.

A set A is open if and only if int(A)=A.
Proof:
→
Obviously int(A)⊆A because if a point is outside the set, then all neighborhoods would contain themselves, which is not within the set, and implying that no neighborhood is completely contained within A, and so it is not within its interior. If x∈A, then A itself is an open neighborhood of A contained within A, and so is within int(A).
←
int(A)=A. Thus, any element of A must have a neighborhood contained within A. The union of all of those neighborhoods is A, and so A is open.

A set A is closed if and only if Cl(A)=A.
Proof:
→
A is closed. Clearly A⊆Cl(A). If x∈Cl(A), then all neighborhoods of X must meet A, meaning that no neighborhood is completely within X-A, implying that it is not within X-A since X-A is open.
←
Cl(A)=A. Consider any point within X-A. It is not within the closure of A, implying that there is a neighborhood of that point completely contained within X-A. Thus, X-A is open, and so A is closed.

Cl(Cl(A)) = Cl(A) (i. e. the closure of a set is closed)
Proof:
Clearly Cl(A)⊆Cl(Cl(A)). All points not within Cl(Cl(A)) must have a neighborhood not meeting Cl(A), and thus not meeting A (because A⊆Cl(A), and so it would not be within Cl(A).

Exercises

1. Prove the following identities for subsets $A,B$ of a topological space $X$:
   • $\mathrm{C}\mathrm{l}(A\cup B)=\mathrm{C}\mathrm{l}(A)\cup \mathrm{C}\mathrm{l}(B)$
   • $\mathrm{C}\mathrm{l}(A\cap B)\subseteq \mathrm{C}\mathrm{l}(A)\cap \mathrm{C}\mathrm{l}(B)$
   • $\mathrm{I}\mathrm{n}\mathrm{t}(A)\cup \mathrm{I}\mathrm{n}\mathrm{t}(B)\subseteq \mathrm{I}\mathrm{n}\mathrm{t}(A\cup B)$
   • $\mathrm{I}\mathrm{n}\mathrm{t}(A\cap B)=\mathrm{I}\mathrm{n}\mathrm{t}(A)\cap \mathrm{I}\mathrm{n}\mathrm{t}(B)$
2. Show that the following identities need not hold (i.e. give an exaple of a topological space and sets $A$ and $B$ for which they fail):
   • $\mathrm{C}\mathrm{l}(A\cap B)=\mathrm{C}\mathrm{l}(A)\cap \mathrm{C}\mathrm{l}(B)$
   • $\mathrm{I}\mathrm{n}\mathrm{t}(A)\cup \mathrm{I}\mathrm{n}\mathrm{t}(B)=\mathrm{I}\mathrm{n}\mathrm{t}(A\cup B)$