Problem 9 - Analysis Now

Question

A subset \(Y\) of a topological space \((X, \tau)\) is an \(F_{\sigma}\)-set, if \(Y=\bigcup F_{n}\), where \(\left(F_{n}\right)\) is a sequence of closed subsets of \(X\). Show that every \(F_{\sigma}\)-set in a normal space is normal (in the relative topology). Hint: Assume first that \(G\) is an open \(F_{\sigma}\)-set of the form \(G=\bigcup G_{n}\), where each \(G_{n}\) is open and \(G_{n}^{-} \subset G_{n+1} .\) Let \(f: C \rightarrow[0,1]\) be a continuous function on a relatively closed subset \(C\) of \(G\). Construct inductively continuous functions \(f_{n}\) on \(\left(G_{n}^{-} \cap G\right) \cup G_{n-1}^{-}\)such that \(f_{n}\left|G_{n-1}^{-}=f_{n-1}\right| G_{n-1}^{-}\)and \(f_{n}\left|G_{n}^{-} \cap C=f\right| G_{n}^{-} \cap C\). Then \(\left(f_{n}\right)\) defines a unique, continuous extension of \(f\), which proves that \(G\) is normal. In the general case \(Y=\bigcup F_{n}\), let \(E\) and \(F\) be closed subsets of \(X\) such that \((E \cap Y) \cap(F \cap Y)=\emptyset\). Thus, \(H=X \backslash(E \cap F)\) is open and \(Y \subset H\). Construct inductively open subsets \(G_{n}\) such that \(F_{n} \bigcup G_{n}^{-} \subset\) \(G_{n+1} \subset H\). Then \(G=\bigcup G_{n}\) is open and normal, so \(E \cap G\) and \(F \cap G\) can be separated by open, disjoints subsets \(A\) and \(B\) in \(G\). Consequently, \(A \cap Y\) and \(B \cap Y\) are relatively open subsets of \(Y\) that separate \(E \cap Y\) and \(F \cap Y\).

Step-by-Step Solution

Answer

Every \( F_{\sigma} \)-set in a normal space is normal because open \( F_{\sigma} \)-sets can be constructed to be normal, allowing separation of closed sets within any \( F_{\sigma} \)-set.

Step 1: Understand the Definitions

An \( F_{\sigma} \)-set is a countable union of closed sets in a topological space. A normal space is one where any two disjoint closed sets can be separated by disjoint open sets. Our task is to show such a set in a normal space is also normal in its relative topology.

Step 2: Consider a Special Case

Assume \( G \) is an open \( F_{\sigma} \)-set given by \( G = \bigcup G_n \) with \( G_n \) open and \( \overline{G_n} \subset G_{n+1} \). We proceed by considering a continuous function \( f: C \rightarrow [0,1] \) where \( C \) is a relatively closed subset of \( G \).

Step 3: Constructing Functions

Inductively define continuous functions \( f_n \) such that \( f_n|_{G_{n-1}^{-}} = f_{n-1}|_{G_{n-1}^{-}} \) and \( f_n|_{G_n^{-} \cap C} = f|_{G_n^{-} \cap C} \). This implies continuity between each successive \( f_n \) on the required domains.

Step 4: Ensuring a Continuous Extension

The sequence \( (f_n) \) defines a unique continuous function that continuously extends \( f \) over \( G \). This indicates \( G \) is indeed normal based on the relative topology.

Step 5: Generalizing to Any \( F_{\sigma} \)-set in Normal Space

Consider \( Y = \bigcup F_n \), where you need to separate \( E \cap Y \) and \( F \cap Y \). Assume \( E \) and \( F \) are closed in the ambient normal space \( X \) with \((E \cap Y) \cap (F \cap Y) = \emptyset\). Construct \( H = X \setminus (E \cap F) \), which is open.

Step 6: Inductive Construction of Open Sets

Define open sets \( G_n \) such that \( F_n \cup \overline{G_n} \subset G_{n+1} \subset H \). This setup ensures that all properties required to form an \( F_{\sigma} \)-set are maintained.

Step 7: Final Normality Proof

\( G = \bigcup G_n \) is open and possessing a normality property, which allows separation of \( E \cap G \) and \( F \cap G \) by disjoint open subsets \( A \) and \( B \). Consequently, \( A \cap Y \) and \( B \cap Y \) separate \( E \cap Y \) and \( F \cap Y \) in \( Y \). Thus, \( Y \) is normal.

Key Concepts

F_sigma Sets

An \( F_{\sigma} \)-set is a fundamental concept in topology, representing a specific category of sets within topological spaces. These are countable unions of closed sets, an attribute that provides them with particular properties useful in analysis and topology. Imagine a sequence of closed sets, \( F_1, F_2, F_3, \ldots \), each of which is closed in the topological space \( X \). An \( F_{\sigma} \)-set is constructed as \( Y = \bigcup_{n=1}^\infty F_n \).

This structure allows \( F_{\sigma} \)-sets to retain some closed-like properties while potentially being open depending on how the sequence is arranged. The ability to express a set in terms of a sequence of closed sets offers flexibility in various mathematical proofs and applications, particularly in normal spaces, as seen in this exercise.

Normal Spaces

Normal spaces are a class of topological spaces that possess a certain desirable property: the ability to separate disjoint closed sets with disjoint open neighborhoods. This is formally stated as: given two disjoint closed sets \( E \) and \( F \), there exist open sets \( U \) and \( V \) such that \( E \subseteq U \), \( F \subseteq V \), and \( U \cap V = \emptyset \).

This separation property is crucial for proving results and constructing functions in analysis. In the context of \( F_{\sigma} \)-sets, this property becomes vital when demonstrating that such a set is also normal in the relative topology, allowing the extension and separation of functions continuously and smoothly.

In this exercise, the normality of the topological space \( X \) permits the use of relative topology on an \( F_{\sigma} \)-set within \( X \), guaranteeing that the space behaves intuitively when considering open and closed subsets.

Continuous Functions

Continuous functions are mappings between topological spaces that preserve the notion of closeness. Formally, a function \( f: X \to Y \) is continuous if for any open set \( V \) in \( Y \), the pre-image \( f^{-1}(V) \) is open in \( X \). This continuity assures that there is no abrupt change or jump as you move from one point to another within the domain.

In the exercise, continuous functions are employed to extend a function defined on a relatively closed subset \( C \) of an \( F_{\sigma} \)-set. By constructing a sequence of continuous functions \( f_n \) that agree on overlapping domains, a seamless, unique extension throughout the entire \( F_{\sigma} \)-set is achieved. This technique effectively uses continuity to bridge local and global perspectives in topology.

This feature of continuous functions underscores their essential role in ensuring smooth extensions over complex structures, providing a toolset to handle functions on topological spaces.

Topology Separation Axioms

Separation axioms in topology describe distinctions between various classes of spaces based on the ability to distinguish points and sets with open sets. The focus in this context is on the T1 and T2 (Hausdorff) axioms, among others, which establish conditions under which points and closed sets can be separated by neighborhoods.

The T1 axiom requires that for any two distinct points, there exist neighborhoods that contain one point but not the other. The T2 or Hausdorff axiom goes further, stating that any two distinct points have disjoint neighborhoods.

In this exercise, these axioms underpin the process by which normal spaces are analyzed. Specifically, by structuring the separation of closed sets within the \( F_{\sigma} \)-set, we rely on properties derived from these separation axioms. They ensure that topologies behave well for functions and subsets, allowing more advanced properties, like normality, to hold in more complex topological arrangements.