91Ó°ÊÓ

(Completely normal spaces.) Show that the following conditions on a topological space \((X, \tau)\) are equivalent: (i) Every subset of \(X\) (equipped with the relative topology) is a normal space. (ii) Every open subset of \(X\) is a normal space. (iii) If \(Y\) and \(Z\) are subsets of \(X\) such that \(Y^{-} \cap Z=Y \cap Z^{-}=\emptyset\), they can be separated by open, disjoint sets in \(X\). Hints: For (ii) \(\Rightarrow\) (iii), consider the open, hence normal, subspace \(G=\) \(X \backslash\left(Y^{-} \cap Z^{-}\right)\)and separate \(Y^{-} \cap G\) and \(Z^{-} \cap G\) with open, disjoint sets \(A\) and \(B\). Show that, in fact, \(Y \subset A\) and \(Z \subset B\). For (iii) \(\Rightarrow\) (i) consider a pair \(Y, Z\) of relatively closed, dis joint subsets of an arbitrary subset \(H\) of \(X\). Show that \(Y^{-} \cap Z=Z \cap Y^{-}=\emptyset\).

Step by step solution

01

Understanding the Implications

To show the equivalence, we want to demonstrate that (i) implies (ii), (ii) implies (iii), and (iii) implies (i). Start by analyzing these implications.

02

Proving (i) implies (ii)

Assume every subset of \(X\) is a normal space, especially any open subset \(U\). Since \(U\) is itself a subset of \(X\), it must be normal as per the assumption in (i). Therefore, every open subset \(U\) of \(X\) is normal, which proves (ii).

03

Proving (ii) implies (iii)

Assume that every open subset of \(X\) is normal. Given subsets \(Y\) and \(Z\) of \(X\) with \(Y^{-} \cap Z = Z^{-} \cap Y = \emptyset\), choose the open set \(G = X \backslash (Y^{-} \cap Z^{-})\). Since \(G\) is open and normal, separate \(Y^{-} \cap G\) and \(Z^{-} \cap G\) using open, disjoint sets \(A\) and \(B\) in \(G\). Then, \(Y \subseteq A\) and \(Z \subseteq B\), thus proving (iii).

04

Proving (iii) implies (i)

Assume for any subsets \(Y\) and \(Z\) where \(Y^{-} \cap Z = Z^{-} \cap Y = \emptyset\), they can be separated by open sets in \(X\). For any subset \(H\) of \(X\), let \(Y\) and \(Z\) be disjoint closed subsets in \(H\). Then \(Y^{-H} \cap Z \subseteq Y^{-X} \cap Z = \emptyset\) and \(Z^{-H} \cap Y \subseteq Z^{-X} \cap Y = \emptyset\), which can be separated by open sets in \(X\), showing that \(H\) is a normal space as it satisfies the normal separation condition, thus proving (i).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Spaces

Normal spaces are a fundamental concept in topology. They are topological spaces where, for any two disjoint closed sets, there exist two disjoint open sets containing each closed set. This property is essential in ensuring certain separation properties in topology.

To look at them more deeply:

• A space is normal if for any closed sets \(F\) and \(G\) with \(F \cap G = \emptyset\), there exist open sets \(U\) and \(V\) so that \(F \subset U\), \(G \subset V\) and \(U \cap V = \emptyset\).
• This condition helps in proving the Urysohn's lemma and Tietze extension theorem, both significant in topology.

Understanding normal spaces is crucial because they often appear as intermediate steps in proving topological properties of other spaces.

Open Subsets

Open subsets in topology are critically important as they form the building blocks of any topological space. An open subset is a set that, intuitively, does not include its boundary.

• An open subset \(U\) of a topological space \((X, \tau)\) is, by definition, a member of the topology \(\tau\).
• For every point \(x\) in \(U\), there is always a little `wiggle room` around \(x\), meaning a neighborhood entirely contained in \(U\).

Open subsets also help in defining and understanding the concept of continuity in functions between topological spaces.

In exploring normal spaces, considering open subsets is vital because these subsets might themselves possess normality which is an important condition for proving equivalence in topology.

Relative Topology

The relative topology, also known as the subspace topology, is a way to construct a topology on any subset of a topological space. It lets you treat a subset as a topological space in its own right.

Here's a quick guide:

• If \((X, \tau)\) is a topological space and \(Y\subseteq X\), the relative topology \(\tau_Y\) on \(Y\) includes sets \(U \cap Y\), where \(U \in \tau\).
• It makes the subset \(Y\) a topological space by itself, inheriting its properties from \(X\).

This concept is crucial when working through problems that involve examining properties like normality at a more localized level within spaces.

Step-by-Step Proof

A step-by-step proof is a detailed breakdown of logical steps taken to prove a particular mathematical statement or theorem.

In topology, such proofs are invaluable:

• They help in understanding the logical flow of arguments. Every step is a small, logical leap from the previous one.
• By explaining implications like (i) implies (ii), (ii) implies (iii), and (iii) implies (i), they reveal how different conditions are interconnected.

For instance, in verifying that subsets in normal spaces also satisfy normal conditions, each proof step shows how specific properties hold under certain assumptions, contributing to a comprehensive understanding of the concepts involved.