Definition and properties

Closed sets

1. Closed sets

   Let $X$ be a space. We say that $A$ is closed if $X-A$ is open.

2. Remark

   Unless we explicitly say otherwise, $\mathbb{R}$ will denote the space of the real numbers with usual topology.

3. Examples

   • Any interval of the form $[a,b]\subseteq X$, where $X$ has the order topology, is closed.
   • The set $[0,1)\subseteq \mathbb{R}$ is not closed (nor open).
   • In a discrete topology, any set is closed.
   • In a cofinite topology on $X$, exactly $X$ and their finite subsets are closed.

Properties of closed sets

1. Properties of closed sets

   Let $X$ be a space. Then:

   • (C1): $\emptyset$ and $X$ are closed.
   • (C2): If $A_{\alpha}$ is closed for each $\alpha\in I$, then $\cap_{\alpha\in I} A_{\alpha}$ is closed.
   • (C3): If $A_1, A_2$ are closed, then $A_{1}\cup A_2$ is closed.

2. Proof

   Exercise.

3. Remark

   We can prove that if a collection of subsets of $X$ satisfies the previous conditions, then their complements form a topology.

Properties of closed sets

1. Closed in subspaces

   Let $Y$ be a subspace of $X$. Then $A\subseteq Y$ is closed in $Y$ if and only if there is $C\subseteq X$ closed in $X$ such that $A=C\cap Y$.

2. Proof
   • Let $A$ closed in $Y$. Then $Y-A$ is open in $Y$, so there is $U$ open in $X$ with $Y-A=Y\cap U$.
   • We prove then that $A=(X-U)\cap Y$.
   • The proof of the converse is an exercise. $\Box$

3. Closed in closed

   Let $Y$ be a subspace of $X$. If $Y$ is closed in $X$ and $A\subseteq Y$ is closed in $Y$, then $A$ is closed in $X$.

Links

• Closed set - Wikipedia, the free encyclopedia

Exercises

• Is there a nondiscrete space where the open sets are the same as the closed sets?
• Show that if $A$ is closed in $X$ and $B$ is closed in $Y$, then $A\times B$ is closed in $X\times Y$.