Lusin spaces are strongly Lindelöf
Our goal is to prove that a Lusin space is strongly Lindelöf. We will need the elementary corollary below to prove the key proposition.
Lemma. Let \(E\) be a set and \(\mathcal{E}\subset\mathrm{Pow}(E)\). Then the \(\sigma\)-field \(\sigma(\mathcal{E})\) is the smallest collection of subsets \(E\) containing \(\varnothing\), \(\mathcal{E}\), \(\{E\setminus A \mid A \in \mathcal{E}\}\) and closed under countable intersections and countable disjoint unions.
Proof. Let \(\mathcal{R}\) be the smallest collection of subsets \(E\) containing \(\varnothing\), \(\mathcal{E}\), \(\{E\setminus A \mid A \in \mathcal{E}\}\) and closed under countable intersections and countable disjoint unions. Let \(\mathcal{R}' = \{A \in \mathcal{R} \mid E \setminus A \in \mathcal{R}\}\). Then \(\mathcal{E} \subset \mathcal{R}'\), and \(\mathcal{R}'\) is closed under complements. So it is enough to show that \(\mathcal{R}'\) is closed under countable unions. But \(\bigcup_n A_n = \bigcup_n (A_n \setminus \bigcup_{i<n}A_i)\), so it is sufficient to show that \(\mathcal{R}'\) is closed under finite unions. Let \(A,B \in \mathcal{R}'\). One has \(A \cup B = (A \setminus B) \cup (B \setminus A) \cup (A \cap B)\) and this is a disjoint union, so \(A \cup B \in \mathcal{R}\). Moreover \(E \setminus (A \cup B) = (E \setminus A) \cap (E \setminus B) \in \mathcal{R}\), hence \(A \cup B \in \mathcal{R}'\). □
Proposition. Let \(E\) be a topological space. A class of Borel sets of \(E\) is equal to yhe Borel \(\sigma\)-field \(\mathcal{B}_E\) whenever it contains all open sets and closed sets and it is closed under countable intersections and countable disjoint unions.
Proof. Let \(\mathcal{C} \subset \mathcal{B}_E\) be such a class. Let \(\mathcal{E}\) be the collection of open sets of \(E\), so that \(\mathcal{B}_E = \sigma(\mathcal{E})\). Obviously one has \(\varnothing \in \mathcal{C}\). Moreover, \(\mathcal{C}\) contains \(\mathcal{E}\) and \(\{E\setminus A \mid A \in \mathcal{E}\}\). Therefore, by the lemma above, \(\mathcal{C}\) contains \(\sigma(\mathcal{E})\). □
Corollary. Let \(E\) be a metric space. A class of Borel sets of \(E\) is equal to \(\mathcal{B}_E\) whenever it contains all open sets and it is closed under countable intersections and disjoint countable unions.
Proof. A closed set in a metric space is the countable intersection of some open sets. Therefore, if \(\mathcal{C}\subset\mathcal{B}_E\) contains all open sets and is closed under countable intersections, it contains all closed sets. Thus the result is a consequence of the previous proposition. □
Disjoint union topology. We will resort to the disjoint union in the proof of the key proposition below. Let \({\{X_i\}}_{i \in I}\) be a collection of sets. The disjoint union of the \(X_i\) is the set \[ \bigsqcup_{i \in I} X_i = \bigl\{(x,i) \mid i \in I, x \in X_i\bigr\}. \] Define the canonical injections \(\varphi_i \colon X_i \to \bigsqcup_{i \in I} X_i\) by \(\varphi_i(x) = (x,i)\). When the \(X_i\) are topological spaces, the disjoint union topology on \(\bigsqcup_{i \in I} X_i\) is the finest topology on \(\bigsqcup_{i \in I} X_i\) for which the canonical injections are continuous. In other words, this is the final topology with respect to \({\{\varphi_i\}}_{i \in I}\). This topology is, denoting by \(\tau_i\) the topology on \(X_i\), \[ \tau = \bigl\{ U \in \mathrm{Pow}(X) \mid \forall i \in I, \varphi_i^{-1}(U) \in \tau_i\bigr\}. \] If \(I\) is countable and the \(X_i\) are second-countable, then \(\bigsqcup_{i \in I} X_i\) is second-countable. Now assume the \(X_i\) are metrizable, and for each \(i \in I\) take a metric \(d_i \leqslant 1\) on \(X_i\) compatible with the topology of \(X_i\). For \(i,j \in I\), \(x \in X_i\) and \(y \in Y_j\), define \[ d\bigl((x,i), (y,j)\bigr) = \begin{cases} d_i(x,y) & \text{ if } i = j \\ 2 & \text{ if } i \neq j \end{cases}. \] It is easy to check that \(d\) is a metric on \(\bigsqcup_{i \in I} X_i\). Clearly, each \(\varphi_i\) is \((d_i,d)\)-continuous. If \(Y\) is a topological space, and \(g \colon \bigsqcup_{i \in I} X_i \to Y\) is a map such that every \(g \circ \varphi_i\) is continuous, it is easy to see that \(g\) is continuous for \(d\). Thus \(d\) induces the disjoint union topology, by the characteristic property of the final topology. It is easy to see that \(d\) is complete if the \(d_i\) are complete. Since second-countability is equivalent to separability in a metric space, we deduce from the above that the disjoint union of a countable family of Polish spaces is a Polish space.
Key proposition. Let \(E\) be a Polish space and \(B \in \mathcal{B}_E\). There exist a Polish space \(Z\) and a continuous bijection from \(Z\) onto \(B\).
Proof. Let \(\mathcal{C}\) be the collection of sets \(B \subset E\) for which there exist a Polish space and a continuous bijection from \(Z\) onto \(B\). An open subset of a Polish space is Polish, therefore open sets of \(E\) belong to \(\mathcal{C}\). By the previous corollary, it is sufficient to show that \(\mathcal{C}\) is closed under countable intersections and countable disjoint unions. Let \(B_0\), \(B_1\), \(\ldots\) belong to \(\mathcal{C}\). Take Polish spaces \(Z_i\) and continuous bijections \(g_i\colon Z_i \to B_i\). Define \[ Z = \left\{ z \in \prod_{i\geqslant 0} Z_i \mid g_0(z_0) = g_1(z_1) = \cdots \right\}. \] Then \(Z\) is closed in the Polish space \(\prod_{i\geqslant 0} Z_i\), therefore is Polish. Define \(g \colon Z \to E\) by \(g(z) = g_0(z_0)\). Then \(g\) is a continuous bijection from \(Z\) onto \(\bigcap_{i \geqslant 0} B_i\). Thus \(\mathcal{C}\) is closed under countable intersections.
Now let \(B_0\), \(B_1\), \(\ldots\) be pairwise disjoints sets belonging to \(\mathcal{C}\). Take Polish spaces \(Z_i\) and continuous bijections \(g_i\colon Z_i \to B_i\). The disjoint union \(Z = \bigsqcup_{i \geqslant 0}Z_i\) of the \(Z_i\) is a Polish space. Define \(g\colon Z \to E\) by \(g\bigl((z,i)\bigr) = g_i(z)\). Then \(g\) is a continuous bijection from \(Z\) onto \(\bigcup_{i \geqslant 0} B_i\). Thus \(\mathcal{C}\) is closed under countable disjoint unions. □
Say that a topological space is a Lusin space if it is homeomorphic to a Borel set of a compact metric space.
Corollary. If \(E\) is a Lusin space, then there exist a Polish space \(Z\) and a continuous bijection from \(Z\) onto \(E\).
Proof. A compact metric space is Polish. Apply the previous proposition. □
To prove the proposition below, we use the results of the first section of the previous post.
A topological space is said to be strongly Lindelöf if every open set of this space is Lindelöf.
Proposition. A Polish space is strongly Lindelöf.
Proof. A Polish space is metrizable and separable, therefore it is Lindelöf. Since an open subset of a Polish space is Polish, we see that a Polish space is strongly Lindelöf. □
Lemma. The continuous image of a strongly Lindelöf space is a strongly Lindelöf space.
Proof. Let \(E_1\) and \(E_2\) be two topological spaces with \(E_1\) strongly Lindelöf, and let \(f\colon E_1 \to E_2\) be a continuous function. Let \(O_2 \subset E_2\) be an open subset of \(f(E_1)\) and let \(\mathcal{C}_2\) be an open cover of \(O_2\). Then \(\bigl\{f^{-1}(C) \mid C \in \mathcal{C}_2\bigr\}\) is an open cover of the open set \(O_1 := f^{-1}(O_2)\). Then it has a countable subcover \(\bigl\{f^{-1}(C) \mid C \in \mathcal{D}_2\bigr\} =: \mathcal{D}_1\) where \(\mathcal{D}_2\) is a countable subset of \(\mathcal{C}_2\). One has \(f\bigl(f^{-1}(C)\bigr) = C\) for every \(C \subset f(E_1)\), therefore \(f(\mathcal{D}_1) = \mathcal{D}_2\) is an open subcover of \(f(O_1) = O_2\). Thus \(f(E_1)\) is strongly Lindelöf. □
Theorem. A Lusin space is strongly Lindelöf.
Proof. Let \(X\) be a Lusin space, and \(f \colon Y \to X\) be a surjective continuous function from a Polish space \(Y\) onto \(X\). By the previous proposition, \(Y\) is strongly Lindelöf, therefore \(X\) is strongly Lindelöf by the previous lemma. □
References
[1] Alexander S. Kechris. Classical descriptive set theory. 1995.
[2] S. M. Srivastava. A Course on Borel Sets. 1998.
[3] Nikolaos S. Papageorgiou, Patrick Winkert. Applied Nonlinear Functional Analysis: An Introduction. 2018.