MATH 190A - Lecture 4

Base

Closed Sets

A set $A\subseteq X$ is closed $⟺$ $X-A$ is open. By DeMorgan, we see that

1. $\varnothing ,X$ are closed
2. Arbitrary intersection of closed sets is closed.

$$\bigcap _{i\in I}{F}_i=\bigcap _{i\in I}(X-U_i)=X-\bigcap _{i\in I}{U}_i$$

3. Finite unions of closed sets are closed.

Remark

Clearly knowing open sets is equivalent to knowing the closed sets. We can define a topology by defining the closed sets satisfying these three axioms.

Interior and Closure

For any set $A\subseteq (X,\tau )$ let

$$A^{\prime }\equiv \text{Int}(A):=\bigcup _{U\text{open},U\subseteq A}U\subseteq A$$

this is clearly an open set $\subseteq A$. Furthermore $\text{Int}(A)=A⟺A$ is open.

Alternatively, this is the “largest” open set inside $A$ (with the notion of inclusion).

Similarly

$$\overline{A}\equiv \text{Cl}(A):=\bigcap _{F\text{ closed},F\supseteq A}F\supseteq A$$

this is a closed set containing $A$.

1. $A$ is closed $⟺\overline{A}=A$
2. Alternatively, this is the “smallest” closed set $\supseteq A$.

It is a consequence of DeMorgan that

$$\overline{A}=X-(X-A{)}^{\prime }$$

Note that

$$\text{Int}(A)=\{x\in A:\exists \text{ open set }U,x\in U\subseteq A\}$$

Hence

$$\begin{array}{rl}x\in \overline{A}& ⟺x\notin \text{Int}(X-A)\\ & ⟺\nexists \text{ open }U\text{ containing }x\text{ inside }X-A\\ & ⟺\forall \text{ open }U\text{ containing }x,U⊈X-A\\ & ⟺\forall \text{ open }U\text{ containing }x,U\cap A\ne \varnothing \end{array}$$