Problem 13 - Analysis Now

Question

(Connected spaces.) A topological space \((X, \tau)\) is connected if it cannot be decomposed as a union of two nonempty disjoint open sets. A subset of \(X\) is clopen if it is both open and closed. Show that \(X\) is connected iff \(\emptyset\) and \(X\) are the only clopen subsets. Let \(f: X \rightarrow Y\) be a surjective continuous map between topological spaces. Show that \(Y\) is connected if \(X\) is.

Step-by-Step Solution

Answer

\(X\) is connected iff the only clopen subsets are \(\emptyset\) and \(X\); if \(X\) is connected, \(Y\) is connected for surjective continuous \(f: X \to Y\).

Step 1: Understanding Clopen Sets in Connected Spaces

By definition, a space \(X\) is connected if it cannot be partitioned into two nonempty disjoint open sets. Thus, if a subset \(U\) is clopen (both open and closed) and \(U eq \emptyset\) and \(U eq X\), then \(X\) can be written as a disjoint union \(U \cup (X \setminus U)\), contradicting connectivity of \(X\). Hence, in a connected space, the only clopen subsets are \(\emptyset\) and \(X\) itself.

Step 2: Proving If Only Clopen Sets Are \(\emptyset\) and \(X\), Then \(X\) Is Connected

Assume that the only clopen sets in \(X\) are \(\emptyset\) and \(X\). Suppose for contradiction that \(X\) is not connected. Thus, there exist nonempty disjoint open sets \(U\) and \(V\) such that \(X = U \cup V\). Both \(U\) and \(V\) would have to be clopen (as their complements are open, making them closed), contradicting our assumption. Hence, \(X\) must be connected.

Step 3: Showing \(Y\) Is Connected When \(f: X \to Y\) Is Surjective and Continuous

Let \(f : X \rightarrow Y\) be continuous and surjective, and assume \(X\) is connected. Assume for contradiction that \(Y\) is not connected, meaning there exist nonempty disjoint open sets \(A\) and \(B\) such that \(Y = A \cup B\). The pre-image of these open sets under \(f\), \(f^{-1}(A)\) and \(f^{-1}(B)\), would be open in \(X\) due to the continuity of \(f\) and nonempty because \(f\) is surjective. But this would mean \(X\) is not connected, giving us a contradiction. Thus, \(Y\) must be connected.

Key Concepts

Topological Spaces

A topological space is a fundamental concept in mathematics. It consists of two parts: a set, typically denoted as \(X\), and a topology \(\tau\) on that set. The topology \(\tau\) is a collection of subsets of \(X\), known as open sets, that satisfies certain properties:

• Both the empty set \(\emptyset\) and the entire set \(X\) are in \(\tau\).
• The intersection of any finite number of sets in \(\tau\) is also in \(\tau\).
• The union of any collection of sets in \(\tau\) is in \(\tau\).

These properties help define the "shape" or "structure" of the space, allowing us to talk about continuity, limits, and other concepts without reference to a specific geometry or metric.

Clopen Sets

Clopen sets are special subsets in a topological space that are simultaneously closed and open. In most topological spaces, few sets have this property. Typically, only the empty set \(\emptyset\) and the entire set \(X\) itself are clopen.This becomes significant in connected spaces, where the presence of clopen sets can determine whether the space is truly connected. If any set other than \(\emptyset\) or \(X\) is clopen, it means the space can be split into disjoint open sets, indicating it is not connected. Thus, in connected spaces, the only clopen subsets are \(\emptyset\) and \(X\). This principle encapsulates the essence of connectedness in topological terms.

Continuous Maps

In topology, a continuous map refers to a function \(f: X \rightarrow Y\) between two topological spaces that preserves the "closeness" of points. For a function to be continuous, the pre-image of every open set in \(Y\) must be an open set in \(X\).Continuity helps in transferring the properties of one space to another. If a function is continuous and the space \(X\) is connected, then any attempt to divide the image space \(Y\) into disjoint open sets contradicts the nature of the continuity. Thus, connectedness is preserved, ensuring that \(Y\) remains a single piece, without gaps or separations.

Surjective Functions

A function \(f: X \rightarrow Y\) is called surjective or onto if every element in the set \(Y\) is the image of at least one element in \(X\). Mathematically, this means for each \(y \in Y\), there exists an \(x \in X\) such that \(f(x) = y\).In the context of topological spaces, surjectivity combined with continuity can strongly influence the properties of the spaces involved. If \(X\) is connected and \(f: X \rightarrow Y\) is a continuous and surjective map, the interconnectedness of \(X\) is "projected" onto \(Y\). Essentially, \(Y\) inherits the connected property from \(X\), as any separation would conflict with the map's completeness and coverage, sustaining \(Y\) as a connected space.