---- > [!definition] Definition. ([[Borel set]]) > Let $X$ be a [[topological space]]. The **Borel $\sigma$-algebra** of $X$ is the [[σ-algebra generated by a set collection|smallest]] [[σ-algebra]] on $X$ containing all open subsets of $X$. Its elements are called the **Borel (sub)sets** of $X$. ^definition > [!basicnonexample] Warning. > There is no finite procedure involving countable unions, countable intersections, and complements for constructing the Borel algebra on $X$. (But there are infinite ones.) ^nonexample > [!equivalence] > If the topology on $X$ is [[topology generated by a basis|generated by a]] [[countably infinite|countable]] [[basis for a topology|basis]] (i.e. is [[second-countable space|second-countable]]), it's enough define the Borel algebra of $X$ to be the smallest $\sigma$-algebra containing the [[basis for a topology|basic open sets]] (e.g. the [[open interval|open intervals]] if $X=\mathbb{R}$ with the [[standard topology on the real line|standard topology]]). ^equivalence > [!basicexample] > - Every [[closed set|closed subset]] of $X$ is a Borel set, being the complement of an open subset. > - Every countable subset $\{ x_{1},x_{2},\dots \}$ of a [[T1 Axiom|T1 space]] is a Borel set, being the countable union of closed sets (singletons) $\{ x_{i} \}$. For example, every countable subset of $\mathbb{R}$ is a Borel set > - "Any subset of $\mathbb{R}$ you can write down in a concrete fashion is a Borel subset" > - Every [[half-open interval]] $[a,b)$ of $\mathbb{R}$ is a Borel set, since it equals the intersection $(a-59, b) \cap [a,b]$ of Borel sets. (Or since $[a,b)=\bigcap_{k=1}^{\infty}\left( a-\frac{1}{k}, b \right)$) > - If $X,Y$ are [[metrizable]] and $f:X \to Y$ a function, then then the set of points at which $f$ is [[continuous]] is the intersection of a sequence of open sets, and thus is a Borel set > > > [!proof] > ^proof Denote by $d$ the [[metric]] [[metric topology|inducing]] the [[topological space|topology]] on $X$ and (abusing notation, we aren't assuming it is the same metric for both spaces) $Y$. For $k \in \mathbb{N}$, let $G_{k}:=\left\{ a \in X: \ex \delta >0 \text{ s.t. } d\big( f(b), f(c) \big) < \frac{1}{k} \text{ for all } b,c \in B_{\delta}(a) \right\}.$ (This condition should feel a bit like if we 'graded' $\varepsilon$ in [[metric-space continuity]] as $\frac{1}{k}$.) $G_{k}$ is an open subset of $X$: take $a \in G_{k}$. Then there is $\delta>0$ such for all $b,c \in B_{\delta}(a)$, $d\big( f(b), f(c) \big) < \frac{1}{k}$. We want to find $\varepsilon>0$ such that $B_{\varepsilon}(a) \subset G_{k}$, i.e., such that given $x \in B_{\varepsilon}(a)$ and $y,z \in B_{\varepsilon }(x)$, there exists $\delta_{x}>0$ such that for all $y,z \in B_{\delta_{x}}(x)$, $d\big( f(y), f(z) \big)< \frac{1}{k}$. So let $x \in B_{\varepsilon}(a)$, with $\varepsilon$ to be determined. If $y \in B_{\delta_{x}}(x)$, $\delta_{x}$ to be determined, then by the triangle inequality $d(y,a) \leq \underbrace{ d(y, x) }_{ < \delta_{x} }+ \underbrace{ d(x, a) }_{ < \varepsilon } < \delta_{x}+\varepsilon$ for all $y \in B_{\delta_{x}}(x)$. Thus, choosing $\delta_{x}=\varepsilon=\frac{\delta}{2}$, $y \in B_{\delta}(a)$. Likewise any $z \in B_{\delta_{x}}(x)$ lives in $B_{\delta}(a)$. This finishes. Note that $\bigcap_{k=1}^{\infty} G_{k}$ equals the set of points at which $f$ is [[continuous]]. Indeed, suppose $f$ is [[continuous]] at $x$. Then $x \in G_{k}$ for all $k$; just take $\varepsilon=\frac{1}{k}$ in the continuity definition to obtaining the $\delta$ witnessing this. On the other hand, if $f$ is not continuous at $x$, then can find $\varepsilon>0$ such that no such $\delta>0$ exists. If we take $K \in \mathbb{N}$ with $K> \varepsilon$, then, we see $x \not \in G_{K}$. Thus, the set of points at which $f$ is [[continuous]] is a Borel set as a countable intersection thereof. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```