A Look into Compactness and Connectedness

Article: "Compact and Connected Spaces" by Richard Johnsonbaugh in Mathematics Magazine

# Article Summary

This article shows how compact and connected spaces are similar, and how, through their similarities, we can characterize spaces that are simultaneously compact and connected. Below is the Theorem and Proof by the author.

Theorem. A space $\mathcal{X}$ is compact and connected if and only if every weakly finitely additive open cover of $\mathcal{X}$ contains $\mathcal{X}$ .

Proof. The sufficiency of the coalition is a straightforward consequence of our previous remarks. So suppose that $\mathcal{X}$ is compact and connected. Let $\mathcal{U}$ be a weakly, finitely additive open cover of $\mathcal{X}$ . We may assume that each member of U is non-void. Since $\mathcal{X}$ is compact, there exists a finite subcover $\{U_1,...,U_n\}$ of $\mathcal{U}$ . We argue by induction on $n$ that $\mathcal{X} \in \mathcal{U}$. The case $n=1$ is obvious. Suppose the statement is true for $n-1$ and that we have a finite subcover $\{U_1,...,U_n\}$ of $\mathcal{U}$. Now $U_1$ meets $U_i$ for some $i$, for otherwise $\{U_1, U_2 \cup...\cup U_n\}$ would form a separation of the connected space $\mathcal{X}$ . We may assume $i=2$. Since $U_1$ meets $U_2$, $U_1 \cup U_2 \in \mathcal{U}$. Thus $\{U_1 \cup U_2, U_3,..., U_n\}$ is a subcover of $\mathcal{U}$ containing $n-1$ sets. By induction $\mathcal{X} \in\mathcal{U}$.

The author reaches this conclusion using the "strongly finitely additive" property of open covers over a compact space, similarly the "weakly infinitely additive" property of open covers over a connected space. With these two, the author finds that the characterizations of compact and connected spaces are "weakly finitely additive." This is how the author is able to prove his theorem.

# Connection to Real Analysis

In real analysis, we study and analyze both compact and connected spaces. However, our text introduces us to the basics of a space as either compact or connected, but not as both simultaneously. We know that compactness is the “most important concept in real analysis” (Pugh 76). It is important because it is able to “reduce the infinite to the finite” (Pugh 76).

Simply stated, “a subset A of a metric space M is compact if every sequence $(a_n)$ in A has a subsequence $(a_n_k)$ that converges to a limit in A” (Pugh 76).

We also know how to determine connectedness; “a metric space M is connected if it is not disconnected – it contains no proper clopen subset” (Pugh 83).