Problem 14 - A History of Mathematics: An Introduction

Question

Show that if \(E\) is a topological space and \(A\) a closed subset, that is, one containing all its accumulation points, then \(E-A\) is a domain (open set) in the notion of Hausdorff. Conversely, if the subset \(B\) of \(E\) is an open set, show that \(E-B\) is closed, using Hausdorff's definition.

Step-by-Step Solution

Answer

Question: Prove that in a topological space E, if A is a closed set, then E-A is an open set, and if B is an open set, then E-B is a closed set. Answer: We have proven: (1) If A is a closed set in E, meaning it contains all its accumulation points, then E-A is an open set due to its complement being closed. (2) If B is an open set in E, then its complement E-B is closed as it contains all its accumulation points. This follows from the Hausdorff's definition of closed and open sets in a topological space.

Step 1: Claim 1: If A is a closed set, then E-A is open.

Given that A is a closed set in the topological space E, meaning A contains all its accumulation points. Let's denote the accumulation points of A by Acc(A). We want to prove that E-A is an open set. By definition of a closed set, A is the closure of itself, which means \(A=\overline{A}=A \cup Acc(A)\). Since \(E-A\) is the complement of A in E, we can write \(E-A=E-(A \cup Acc(A))\). Given E as a topological space, we know that a set is open if its complement is closed. Thus, we need to show that \(E-(E-A)=A \cup Acc(A)\) is closed, which we already know since this is the definition of our closed set A in a Hausdorff space. This means that \(E-A\) is an open set. Claim 1 is proven.

Step 2: Claim 2: If B is an open set, then E-B is closed.

Given that B is an open set in the topological space E, we want to prove that E-B is a closed set. In a Hausdorff space, a set is closed if it contains all its accumulation points. Firstly, let's denote the set of accumulation points of E-B as Acc(E-B). We need to show that E-B contains all of its accumulation points, i.e., \(E-B \subseteq E-B \cup Acc(E-B)\). Consider a point x in E-B, which means \(x \notin B\). Since B is an open set, its complement E-B is closed by definition. Therefore, x must lie in the closure \(\overline{E-B}\). This means that x is either an element of E-B or an accumulation point of E-B. Since x is an arbitrary point in E-B, we can say that all points in E-B lie in \(E-B \cup Acc(E-B)\). This implies that E-B is closed as it includes all its accumulation points. Claim 2 is proven. In conclusion, we have shown that if A is a closed set in a topological space E, then E-A is an open set, or if B is an open set in E, then E-B is a closed set, using the Hausdorff's definition of a closed and open set.

Key Concepts

Closed Sets

In topology, closed sets are an important concept. To understand closed sets, first consider the idea of accumulation points. An accumulation point of a set is a point where every neighborhood of it contains a point of the set different from itself. Now, a set is closed if it contains all its accumulation points. This makes closed sets very inclusive.

Closed sets in a topological space have a specific feature. When you take any space and remove a closed set, the resulting set in the space is open. This will become useful when exploring how closed and open sets relate in different topological contexts, such as Hausdorff spaces.

Understanding closed sets is crucial when discussing complements in a topological space. The set complement of a closed set is what we usually would call an open set. This is a fundamental aspect of the duality between open and closed sets.

Open Sets

Open sets are among the fundamental building blocks of topology. For a set to be open in a topological space, think about it not containing its boundary points. Imagine a space that gives you some breathing room. In mathematical terms, this means that for every point in the open set, there is some neighborhood completely contained in the set.

Open sets are defined by their complement being closed. This means that in a topological space, if you remove an open set, what remains is closed. These notions of 'open' and 'closed' are intrinsically linked, providing a dual perspective of looking at spaces.

• Open sets give you flexibility. You can move a bit without stepping outside of the set.
• They form the basis for creating topologies by describing how spaces are knit together.

Understanding open sets helps in grasping the structure of the larger space, and how the space behaves under various mathematical operations like union and intersection.

Hausdorff Definition

The Hausdorff condition is a fundamental separation axiom in topology. To understand it, consider the idea of separating points. In a Hausdorff space, any two distinct points can be separated by open sets. This property ensures that not only can we distinguish points, but they can also be completely isolated using open neighborhoods.

Using the Hausdorff definition, we can explore how closed and open sets behave in such spaces. In a Hausdorff space:

• A closed set contains all its accumulation points.
• For any open set, removing it results in a closed set.

These help us understand properties like convergence, compactness, and continuity in Hausdorff spaces. Recognizing the Hausdorff condition makes it easier to apply topological concepts to real-world scenarios, as it aligns closely with our intuitive understanding of continuity and separation.