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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Polish space} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{introduction}{Introduction}\dotfill \pageref*{introduction} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{universal_polish_spaces}{``Universal'' Polish spaces}\dotfill \pageref*{universal_polish_spaces} \linebreak \noindent\hyperlink{cantorbendixson_rank}{Cantor-Bendixson rank}\dotfill \pageref*{cantorbendixson_rank} \linebreak \noindent\hyperlink{borel_isomorphism_between_polish_spaces}{Borel isomorphism between Polish spaces}\dotfill \pageref*{borel_isomorphism_between_polish_spaces} \linebreak \noindent\hyperlink{further_examples}{Further examples}\dotfill \pageref*{further_examples} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{introduction}{}\subsection*{{Introduction}}\label{introduction} A \textbf{Polish space} is a [[topological space]] that's [[homeomorphism|homeomorphic]] to a [[separable space|separable]] [[complete space|complete]] [[metric space]]. Every [[second countable space|second countable]] [[locally compact space|locally compact]] [[Hausdorff space]] is a Polish space, among others. Polish spaces provide a useful framework for doing [[measure theory]]. As with any topological space, we can take a Polish space and regard it as a [[measurable space]] with its [[sigma-algebra]] of [[Borel set]]s. Then, there is a very nice classification of Polish spaces up to measurable bijection: there is one for each [[countable set|countable]] [[cardinality]], one whose cardinality is that of the [[continuum]], and no others. Why are Polish spaces `not very big'? In other words, why are there none with cardinality exceeding the continuum? As with any separable metric space, it's because any Polish space has a countable dense subset and you can write any point as a limit of a sequence of points in this subset. So, you only need a [[sequence]] of integers to specify any point in a Polish space. More sharply, see Lemma \ref{baire} below. \begin{example} \label{}\hypertarget{}{} The primordial example (and in practice, one of the most convenient) is [[set-theoretic Baire space|Baire space]] $B$, viz. the space of irrational real numbers between $0$ and $1$, with the [[subspace topology]] inherited from the [[real number|real line]]. This is obviously not complete with respect to the metric induced from the real line, but it is homeomorphic to the product space $\mathbb{N}^\mathbb{N}$ via regular [[continued fraction]] expansions, and the latter is metrizable by a complete metric where the distance between two sequences $a = (a_1, a_2, \ldots)$ and $b = (b_1, b_2, \ldots)$ of positive integers is given by the formula $d(a, b) = 1/2^n$ where $n$ is the least integer such that $a_n \neq b_n$. A countable dense subset is given by continued fractions that eventually repeat (quadratic surds). \end{example} The convenience of Baire space is attested to by the fact that in [[descriptive set theory]], a ``real number'' is often taken to mean just a point of Baire space. See also Theorem \ref{Borel} below. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} Much of the material here is adapted from \hyperlink{Marker}{Marker}, who in turn cites the text by Kechris as a main source. \begin{itemize}% \item A countable product $X = \prod_n X_n$ of Polish spaces is Polish. Indeed, each $X_n$ is metrizable by a complete metric $d_n$ that takes values in $[0, 1]$, and then we may define another such complete metric on $X$ by\begin{displaymath} d(f, g) = \sum_n \frac1{2^{n+1}} d_n(f(n), g(n)). \end{displaymath} The metric topology on $X$ coincides with the product topology. \end{itemize} In particular, the [[Hilbert cube]] $[0, 1]^\mathbb{N}$ is Polish. \begin{itemize}% \item A subspace of a Polish space $X$ is Polish iff it is a [[G-delta set]] of $X$. See \hyperlink{Marker}{Marker}, Theorem 1.33. \end{itemize} \hypertarget{universal_polish_spaces}{}\subsubsection*{{``Universal'' Polish spaces}}\label{universal_polish_spaces} \begin{prop} \label{hilbert}\hypertarget{hilbert}{} Any Polish space is homeomorphic to a subspace of the Hilbert cube. \end{prop} In fact, this is true of any [[separable space|separable metrizable space]]. Note that mere separability does not suffice, as there are separable spaces that are not first-countable, such as the Stone-ech compactification of $\N$, and hence cannot be subspaces of any metrizable space. What distinguishes Polish spaces is that they are, up to homeomorphism, precisely the \emph{$G_\delta$ subsets} of the Hilbert cube. \begin{lemma} \label{baire}\hypertarget{baire}{} Every [[inhabited]] Polish space $X$ admits a continuous surjection from Baire space. \end{lemma} \begin{proof} Construct by induction a collection of closed sets (balls) $C_s$ indexed over \emph{finite} sequences $s$ of positive integers, with the following properties: \begin{itemize}% \item For the empty sequence $e$, $C_e = X$; \item For nonempty sequences $s$, $diam(C_s) \leq 1/n$ where $n = length(s)$; \item Letting $(s, k)$ denote the extension of $s$ obtained by appending to $s$ the final element $k$, \begin{displaymath} C_s = \bigcup_{k=0}^\infty C_{(s, k)}; \end{displaymath} \item If $t$ extends $s$, then $center(C_t) \in C_s$. \end{itemize} Then define $f: B \to X$ where for each infinite sequence $a \in \mathbb{N}^\mathbb{N}$, \begin{displaymath} f(a) = \bigcap \{C_s: a\; extends\; s\}. \end{displaymath} One may check that $f$ is continuous and surjective. \end{proof} \hypertarget{cantorbendixson_rank}{}\subsubsection*{{Cantor-Bendixson rank}}\label{cantorbendixson_rank} Let $C \subseteq X$ be a closed subset of a Polish space $X$. The following operation traces back to Cantor's work in Fourier analysis, which in turn led to his study of [[countable ordinal|countable ordinals]] and [[ordinal]] analysis. For a subset $A \subseteq X$, recall that $x \in A$ is a [[limit point]] if $x \in Cl(A \setminus \{x\})$. A point $x \in A$ that is not a limit point of $A$ is called an \emph{isolated point} of $A$. Clearly each isolated point is open relative to $A$, as is therefore the set of isolated points. \begin{defn} \label{}\hypertarget{}{} The \textbf{Cantor-Bendixson derivative} of $C$ is the set $C' \subseteq C$ of limit points relative to $C$. For each ordinal $\alpha$ the iterated derivative $C^\alpha$ is defined by recursion: $C^0 = C$, $C^{\alpha + 1} = (C^\alpha)'$, and $C^\alpha = \bigcap_{\beta \lt \alpha} C^\beta$ if $\alpha$ is a limit ordinal. \end{defn} Since $\beta \lt \alpha$ implies $C^\beta \supseteq C^\alpha$, it is clear that there is a least ordinal $\alpha$ for which $C^\alpha = C^{\alpha + 1}$. This ordinal is called the \emph{Cantor-Bendixson rank} of $C$. A \emph{[[perfect set]]} is a closed set $C$ such that $C = C'$. (Some people insist that a perfect set also be nonempty; we do not.) \begin{prop} \label{cantor}\hypertarget{cantor}{} For each nonempty perfect set $P$ in a Polish space $X$, there is a continuous injection $i: \mathbf{2}^\mathbb{N} \to P$ from [[Cantor space]]. In particular, the cardinality of $P$ is the [[continuum]] $c$. \end{prop} \begin{proof} Let $T$ be the complete infinite binary tree, whose infinite paths from the root correspond to points in Cantor space $\mathbf{2}^\mathbb{N}$. To each node $s$ in $T$ (a finite sequence of $0$`s and $1$'s) we construct by induction an open set $U_s$ with the following properties: \begin{itemize}% \item $U_e = X$ for the empty sequence $e$, \item $\widebar{U_t} \subseteq U_s$ if $s$ is an initial segment of $t$, \item For the two children $(s, 0)$ and $(s, 1)$ of $s$, the sets $U_{(s, 0)}$ and $U_{(s, 1)}$ are disjoint, \item $diam(U_s) \leq 1/n$ where $n$ is the length of (nonempty) $s$, \item $U_s \cap P \neq \emptyset$ for each $s$. \end{itemize} Indeed, if $U_s$ has been constructed, and given $x \in U_s \cap P$, there are (at least!) two points $x_0, x_1 \in U_s \cap P$ since $P$ is perfect. We can easily find disjoint neighborhoods $U_{(s, 0)}, U_{(s, 1)}$ of these points respectively with the required properties. Then for each path $p = (a_0, a_1, \ldots) \in \mathbf{2}^\mathbb{N}$, define \begin{displaymath} i(p) = \bigcap_{s \preceq p} U_s \end{displaymath} where $s \preceq p$ means $s$ is an initial segment of $p$. The intersection consists of a single point because it equals the intersection of a decreasing chain of closed sets with shrinking diameter (thus closing in on a limit of a [[Cauchy sequence]]). The map $i$ is clearly injective by the disjointness of open sets of children, and it is easy to see $i$ is continuous. \end{proof} \begin{theorem} \label{}\hypertarget{}{} For $C$ a closed subset of a Polish space $X$, the Cantor-Bendixson rank is a countable ordinal $\alpha$. The complement $C \setminus C^\alpha$ is at most countable. \end{theorem} \begin{proof} Let $U_i$ be a countable basis of $X$. For each $\beta \lt \alpha$, each point $x \in C^\beta \setminus C^{\beta + 1}$ is an isolated point, so we can find an $U_{i(x)}$ in the basis such that $U_{i(x)} \cap (C^\beta \setminus C^{\beta + 1}) = \{x\}$. It is then clear that $x \mapsto i(x)$ is injective, so each $C^\beta \setminus C^{\beta + 1}$ is countable. Similarly, whenever $C^\beta \setminus C^{\beta + 1}$ is nonempty, we can find a basis element $U_{j(\beta)}$ that isolates one of its points (say $x$), and this same $U_j$ cannot isolate any point of an earlier $C^\gamma \setminus C^{\gamma + 1}$ since $x$ is a limit point of $C^\gamma$. It follows that $\beta \mapsto j(\beta)$ is an injective map, so that $\alpha$ must be a countable ordinal, and the collection $F \coloneqq C \setminus C^\alpha = \bigcup_{\beta \lt \alpha} C^\beta \setminus C^{\beta + 1}$ is (at most) countable. \end{proof} \begin{cor} \label{CH}\hypertarget{CH}{} A closed set in a Polish space is the disjoint union $P \cup F$ of a perfect set $P$ and a finite or countable set $F$. Hence an infinite closed subset in a Polish space has cardinality either $\aleph_0$ or the continuum $c = 2^{\aleph_0}$ ([[continuum hypothesis]] for closed sets). \end{cor} \hypertarget{borel_isomorphism_between_polish_spaces}{}\subsubsection*{{Borel isomorphism between Polish spaces}}\label{borel_isomorphism_between_polish_spaces} Recall that a function $f: X \to Y$ between topological spaces is \emph{Borel} if $f^{-1}(V)$ is a Borel set in $X$ for every open $V$ in $Y$. Topological spaces and Borel functions form a category. \begin{lemma} \label{}\hypertarget{}{} If $A, B$ are countably infinite $T_1$-[[separation axiom|spaces]], then any bijection $f: A \to B$ is a Borel isomorphism. \end{lemma} For, the inverse image $f^{-1}(U)$ of an open set, being countable, is an $F_\sigma$ set. In particular, any two denumerable Polish spaces are Borel isomorphic. \begin{lemma} \label{}\hypertarget{}{} The unit interval $[0, 1]$ and Cantor space $2^\mathbb{N}$ are Borel isomorphic. \end{lemma} \begin{proof} Let $E \subseteq \mathbf{2}^\mathbb{N}$ be the set of $(0, 1)$-sequences that are eventually constant. Then \begin{displaymath} f(a_1, a_2, \ldots) = \sum_n \frac{a_n}{2^n} \end{displaymath} maps $2^\mathbb{N} \setminus E$ homeomorphically onto the space of non-(dyadic rational) numbers in $[0, 1]$. Pick any bijection $g: E \to \{dyadic\; rationals\}$. Then the union of $f$ and $g$ defines a Borel isomorphism $h: \mathbf{2}^\mathbb{N} \to [0, 1]$. \end{proof} \begin{prop} \label{subset}\hypertarget{subset}{} Any Polish space $X$ is Borel isomorphic to a Borel subset of Cantor space. \end{prop} \begin{proof} There is a sequence of maps \begin{displaymath} X \hookrightarrow [0, 1]^\mathbb{N} \stackrel{h^\mathbb{N}}{\to} (\mathbf{2}^{\mathbb{N}})^\mathbb{N} \stackrel{j}{\to} \mathbf{2}^\mathbb{N} \end{displaymath} where the first map is an inclusion of a $G_\delta$ set, the second is induced from a Borel isomorphism $h$, and the third is a homeomorphism. \end{proof} \begin{theorem} \label{Borel}\hypertarget{Borel}{} Any two Polish spaces of the same cardinality are Borel isomorphic. \end{theorem} \begin{proof} It suffices to prove this for uncountable Polish spaces (which have continuum cardinality, as we saw in Corollary \ref{CH}). We show that any such $X$ is Borel isomorphic to Cantor space. By Proposition \ref{subset}, we have an inclusion $i: X \to \mathbf{2}^{\mathbb{N}}$ that maps $X$ Borel isomorphically onto its image, and by Proposition \ref{cantor}, we have an inclusion $j: \mathbf{2}^{\mathbb{N}} \to X$ that maps Cantor space Borel isomorphically (even homeomorphically) onto its image. The rest is just a matter of checking that at least one of the proofs of the [[Cantor-Schroeder-Bernstein theorem]] applies to this Borel context. Indeed, as explained \href{https://ncatlab.org/nlab/show/Cantor-Schroeder-Bernstein+theorem#alt}{here}, we may consider \begin{displaymath} S = \bigcap_{n \geq 0} (\neg \exists_j \neg \exists_i)^n(X) \end{displaymath} Each of the direct image maps $\exists_i$ and $\exists_j$ takes Borel sets to Borel sets, since their inverses $i^{-1}, j^{-1}$ are Borel functions on their domains. So each of the iterates $(\neg \exists_j \neg \exists_i)^n(X)$ is a Borel set and so is their countable intersection $S$. The set $S$ is a fixed point of $\neg \exists_j \neg \exists_i$ and so we construct a Borel isomorphism $h: X \to \mathbf{2}^{\mathbb{N}}$ as \begin{displaymath} \itexarray{ x & \mapsto & i(x) & if\; x \in S \\ & \mapsto & j^{-1}(x) & if\; x \in \neg S } \end{displaymath} following the proof of the [[Cantor-Schroeder-Bernstein theorem]]. \end{proof} For another proof, see theorem 3.1.1 of \hyperlink{Ber}{Berberian}. \hypertarget{further_examples}{}\subsection*{{Further examples}}\label{further_examples} \begin{itemize}% \item The classical $L^p$-[[Lebesgue space|spaces]] for $p \lt \infty$ are Polish spaces. \item For a metric space $X$, let $K(X)$ be the set of nonempty [[compact space|compact]] subsets of $X$, equipped with the [[Hausdorff metric]]. If $X$ is a separable complete metric space, then so is $K(X)$. \item A [[local compactum|locally compact Hausdorff space]] is Polish iff it is second-countable. \item If $X, Y$ are Polish and $X$ is locally compact, then the [[exponentiable space|exponential]] $Y^X$ (the space of continuous maps with the compact-open topology) is also Polish. (See for example A.10 \href{http://www.math.tamu.edu/~kerr/book/appendixA.pdf}{here}.) \item If $X$ is Polish, then the space $P(X)$ of Borel probability measures $\mu$ equipped with the Prokhorov metric is also Polish. Definitions: for $x \in X$ and $A \subset X$ nonempty, put $d(x, A) = \inf \{d(x, a): a \in A\}$, and for $\alpha \gt 0$ define $A_\alpha = \{x \in X: d(x, A) \lt \alpha\}$ and $\emptyset_\alpha = \emptyset$. Define the \emph{Prokhorov metric} $d_P$ by \begin{displaymath} d_P(\mu, \nu) = \inf \{\alpha \gt 0: \forall_{Borel\; A} \mu(A) \lt \nu(A_\alpha) + \alpha \; and \; \nu(A) \lt \mu(A_\alpha) + \alpha\}. \end{displaymath} Then the map $u_X: X \to P(X)$ sending $x$ to the [[Dirac measure]] $\delta_x$ is continuous. If $x_n$ is a countable dense set of $X$, then the set of rational convex combinations $\sum_{i=1}^n q_n \delta_{x_n}$ (with $0 \leq q_n \leq 1$ and $\sum_i q_i = 1$) is a countable dense subset of $P(X)$. More at [[Giry monad]]. \item Spaces of [[structure in model theory|structures]] and [[models]] (in the [[model theory]] sense), and spaces of $n$-[[n-type (model theory)|types]] (again in the model theory sense), quite often provide examples of Polish spaces. For example, if $L$ is a countable language (a countable [[signature (in logic)|signature]]), then the collection of possible $L$-structures $M$ on the countable universe $\mathbb{N}$, topologized by taking as basic opens \begin{displaymath} U_\phi = \{M \in Struct(L): M \models \phi\} \end{displaymath} where $\phi$ is a quantifier-free sentence, is a Polish space homeomorphic to the product space \begin{displaymath} \prod_{relations\; R} \mathbf{2}^{\mathbb{N}^{arity(R)}} \times \prod_{functions\; f} \mathbb{N}^{\mathbb{N}^{arity(f)}} \end{displaymath} (taking constants to be functions of arity $0$ in the signature). \end{itemize} As an example of the last principle, we have a kind of continuum hypothesis for substructures: \begin{prop} \label{}\hypertarget{}{} Let $X$ be a countable structure of a language; then the number of substructures of $X$ is either countable or the continuum. \end{prop} \begin{proof} A subset of $X$ is specified by its [[characteristic function]] $\chi \in 2^X$, where $2^X$ is regarded as a Polish space. In order for the subset \emph{not} to support a substructure, then must be some function symbol $f$ of the language, say of arity $n$, and elements $a_1, \ldots, a_n$ such that $\chi(a_i) = 1$ for all $i$ and $\chi(f_X(a_1, \ldots, a_n)) = 0$. The basic open \begin{displaymath} U_{f; a_1, \ldots, a_n} \coloneqq \{\omega \in 2^X: \; \omega(a_1) = 1, \ldots, \omega(a_n) = 1, \; \omega(f_X(a_1, \ldots, a_n)) = 0\} \end{displaymath} would thus contain $\chi$ and also exclude any substructure; we conclude that the collection of substructures forms a closed subset $C$ of $2^X$. Then the Cantor-Bendixson theorem (i.e., Corollary \ref{CH}) shows that ${|C|}$ is either countable or the continuum. \end{proof} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item \href{http://golem.ph.utexas.edu/category/2008/08/polish_spaces.html}{Polish spaces}, blog discussion, $n$-category caf\'e{}, 2008- \item David Marker, \emph{Descriptive Set Theory}, UIC Course Notes (Fall 2002) (\href{http://homepages.math.uic.edu/~marker/math512/dst.pdf}{pdf}) \end{itemize} \begin{itemize}% \item S.K. Berberian, \emph{Borel spaces}, (\href{https://www.ma.utexas.edu/mp_arc/c/02/02-156.pdf}{pdf}) \end{itemize} \begin{itemize}% \item A.S. Kechris, \emph{Classical descriptive set theory}, Springer-Verlag (1994). \end{itemize} Internal groupoids in Polish spaces are considered in \begin{itemize}% \item Martino Lupini, \emph{Polish groupoids and functorial complexity}, \href{http://arxiv.org/abs/1407.6671}{arxiv/1407.6671} \end{itemize} [[!redirects Polish spaces]] \end{document}