$G_{\delta }$ and $F_{\sigma }$ subsets of a topological space, agreeing with the real-line notion

Let $(X,\mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) and let $A\subseteq X$.

• $A$ is a $G_{\delta }$ set of $X$ when there is a sequence $(V_n{)}_{n\in \mathbb{N}}$ of open subsets of $X$ with $A={\bigcap }_{n\in \mathbb{N}}{V}_n\mathrm{.}$
• $A$ is an $F_{\sigma }$ set of $X$ when there is a sequence $(F_n{)}_{n\in \mathbb{N}}$ of closed subsets of $X$ with $A={\bigcup }_{n\in \mathbb{N}}{F}_n\mathrm{.}$

As everywhere in this library $\mathbb{N}$ contains $0$, so both indexings start at $0$. An at most countable family may always be presented as a sequence (Finite, countably infinite, countable, uncountable): a finite list $V_0,\dots ,V_m$ is extended by $V_n:=V_m$ for $n>m$, which changes neither the intersection nor the union, so nothing is lost by indexing over $\mathbb{N}$.

The two classes are exchanged by complementation. $A$ is $F_{\sigma }$ in $X$ if and only if $X\setminus A$ is $G_{\delta }$ in $X$. If $A={\bigcup }_n{F}_n$ with each $F_n$ closed then $X\setminus A={\bigcap }_n(X\setminus F_n)$ by De Morgan and each $X\setminus F_n$ is open (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison); the converse is the same computation read backwards.

Every open set is $G_{\delta }$ and every closed set is $F_{\sigma }$, by the constant sequence $V_n:=A$, respectively $F_n:=A$. Neither converse holds, and $\mathbb{R}$ with its usual topology already refutes both. The singleton $\{0\}$ is a $G_{\delta }$ that is not open: it is ${\bigcap }_{n\in \mathbb{N}}(-1\mathrm{/}(n+1),1\mathrm{/}(n+1))$, since $0$ lies in every one of those intervals while a real $t\mathrm{\ne }0$ is excluded at some index, the Archimedean property giving a natural $k\ge 1$ with $1\mathrm{/}k<\mathrm{\mid }t\mathrm{\mid }$ and $k$ being a successor $n+1$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$, Every nonzero natural number is a successor, The canonical natural $\iota (n)=n\cdot 1_F$ of a field); and $\{0\}$ is not open because every bounded open interval $(a,b)$ with $a<0<b$ contains the point $b\mathrm{/}2\mathrm{\ne }0$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded, claim 3). Complementing, $\mathbb{R}\setminus \{0\}$ is an $F_{\sigma }$ that is not closed, its complement $\{0\}$ not being open.

The condition that is a real restriction is the other pairing, namely that every closed set be a $G_{\delta }$, equivalently that every open set be an $F_{\sigma }$. That is not automatic in an arbitrary space, and it is exactly the second conjunct of perfect normality later on this page. It must not be confused with the two automatic inclusions above: they hold everywhere and say nothing about a space.

Agreement with the real-line notion, stated because a second notion of the same name would be a defect. $F_{\sigma }$ and $G_{\delta }$ subsets of $\mathbb{R}$ defines $F_{\sigma }$ and $G_{\delta }$ subsets of $\mathbb{R}$ by the same two displayed conditions, with "open" and "closed" read in the sense of Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen. Those two words name the same two collections of subsets of $\mathbb{R}$ as the usual topology of $\mathbb{R}$ does, and the verification is one line of unfolding. Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen calls $U$ open when every $x\in U$ admits $\epsilon >0$ with $N_{\epsilon }(x)\subseteq U$, where $N_{\epsilon }(x)=(x-\epsilon ,x+\epsilon )$ (The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$); The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement calls $U$ open in $(\mathbb{R},d_{\mathbb{R}})$ when every $x\in U$ admits $r>0$ with $B(x,r)\subseteq U$, and $B(x,r)=(x-r,x+r)$ by claim 2 of The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded. The two conditions are therefore the same condition word for word, so the two collections of open subsets of $\mathbb{R}$ are one collection, and hence so are the two collections of closed subsets, each being the complements of the other collection. The usual topology of $\mathbb{R}$ is the metric topology of $d_{\mathbb{R}}$ (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not). Since the two definitions quantify over one collection of open sets and one collection of closed sets, a subset of $\mathbb{R}$ is $G_{\delta }$ in the sense above, for $\mathbb{R}$ with its usual topology, if and only if it is $G_{\delta }$ in the sense of $F_{\sigma }$ and $G_{\delta }$ subsets of $\mathbb{R}$; and likewise for $F_{\sigma }$. There is one notion here, not two, and every statement proved about $F_{\sigma }$ or $G_{\delta }$ subsets of $\mathbb{R}$ elsewhere in this library may be quoted verbatim as a statement about the topological space $\mathbb{R}$.

• The letters. $F$ for ferme with $\sigma$ for somme, $G$ for Gebiet with $\delta$ for Durchschnitt, as $F_{\sigma }$ and $G_{\delta }$ subsets of $\mathbb{R}$ records.

• Neither class is closed under complementation, which is why both names are needed; and neither is a topology, an arbitrary union of $G_{\delta }$ sets being no longer $G_{\delta }$ in general. What is true, and all that is used on this page, is the complementation duality above together with the fact that a finite intersection of $G_{\delta }$ sets and a finite union of $F_{\sigma }$ sets stay in their class, by rearranging a finite array of sequences.

• In a metric space every closed set is $G_{\delta }$. That is proved later on this page from the distance function, and it is the reason every metrizable space is perfectly normal. In a general space it can fail, so it is a genuine hypothesis and not a convenience.