Category theory gives a topological notion of smallness. A set may be dense and still be small in category; the rational numbers are the main example. Nowhere dense sets are sets whose closures contain no nonempty open set. Countable unions of such sets are category I, and sets not of that form are category II. :::definition[Nowhere Dense Set] A subset $A$ of a metric space $(X,d)$ is $nowhere dense$ if $(\overline{A})^\circ=\varnothing$. ::: :::definition[Category I Set] A subset $A\subset X$ is of $category I$ if $A=\bigcup_{n=1}^{\infty}A_n$, where each $A_n$ is nowhere dense. ::: :::definition[Category II Set] A subset of $X$ that is not of category I is called a set of $category II$. ::: :::example[Rationals Are Category I] Since $\mathbb{Q}=\bigcup_{n=1}^{\infty}\{q_n\}$ and each singleton is nowhere dense in $\mathbb{R}$, the rationals are of category I. ::: :::example[Nowhere Dense Sets in a Discrete Space] In a discrete metric space, the only nowhere dense subset is $\varnothing$. ::: :::theorem[Closed Nowhere Dense Set Criterion] Let $E$ be closed in $(X,d)$. Then $E$ is nowhere dense if and only if every nonempty open set $G$ contains an open ball lying in $G\setminus E$. ::: :::proof Given that $E$ is closed. To prove the criterion. If $E$ is nowhere dense, then $E^\circ=\varnothing$. Let $G$ be nonempty and open. Since $G$ cannot be contained in $E$, choose $x\in G\setminus E$. The set $G\cap(X\setminus E)$ is open and contains $x$, so it contains a ball $S(x,r)$. Hence $S(x,r)\subset G\setminus E$. Conversely, suppose every nonempty open $G$ contains a ball in $G\setminus E$. If $E^\circ$ were nonempty, take $G\subset E$ nonempty and open. Then no ball could lie in $G\setminus E$. A contradiction. Hence $E^\circ=\varnothing$, and since $E$ is closed, $E$ is nowhere dense. $\square$ ::: :::solved-problem Show that the Cantor set is nowhere dense in $[0,1]$. ::: :::solution The Cantor set is closed. Every nonempty open interval in $[0,1]$ contains an interval removed at some finite stage of the construction, so no nonempty open interval lies inside the Cantor set. Hence its interior is empty. Therefore it is nowhere dense. $\square$ ::: :::exercise [1] State the main definition from this lesson. [2] State the main theorem from this lesson. [3] Give one example illustrating the theorem. [4] Explain one common mistake related to this topic. ::: :::answer [1] The main definition is the first titled definition in the lesson. [2] The main theorem is the titled theorem proved in the lesson. [3] The examples in the lesson provide standard choices. [4] A common mistake is to ignore the metric or the ambient space when applying the definition. ::: :::faq[Frequently Asked Questions] Q: Why is the ambient metric space important? A: Openness, closedness, closure, and boundary are interpreted relative to the chosen metric space. Q: Can the same set behave differently in different metric spaces? A: Yes. Changing the metric or the ambient space may change topological behaviour. Q: How should students check a definition? A: They should return to balls, closures, and sequences, depending on the context. ::: :::call-to-action[Continue from Category and Nowhere Dense Sets] subtitle: Review the definitions and proofs before moving through the metric topology sequence. button: Next Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/baire-category-theorem-and-applications button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/separable-metric-spaces :::