Connectedness, Uniform Continuity, and Distance on Met. Sps. Review

Connectedness, Uniform Continuity, and Distance on Metric Spaces Review

We will now review some of the recent material presented.

• Recall from the Connected and Disconnected Metric Spaces page that a metric space $(M, d)$ is said to be Disconnected if there exists $A, B \subseteq M$, $A, B \neq \emptyset$ such that $A \cap B = \emptyset$ and:
(1)
\begin{align} M = A \cup B \end{align}
• In other words, a metric space $M$ is disconnected if there exists two nonempty disjoint subsets of $M$ whose union is $M$. We said that a metric space is Connected if it is not disconnected. Furthermore, we said that a subset $S \subseteq M$ is connected/disconnected if the resulting metric subspace $(S, d)$ is connected/disconnected.
• On the Two-Valued Function Criterion for the Disconnectedness of a Metric Space page we looked at a nice criterion for a metric space to be disconnected. We saw that $(M, d)$ is disconnected if and only if there exists a continuous real-valued function $f : M \to \mathbb{R}$ such that $f(M) = \{ 0, 1 \}$. We showed this by defining a function $f$ that maps every element in $A$ to $0$ and every element in $B$ ]] to [[$1$.
• We then looked at a transferable property on the Continuous Functions on Connected Sets of Metric Spaces page. We saw that if $(S, d_S)$ and $(T, d_T)$ are metric spaces with $C \subseteq S$ and $f : C \to T$ then if $C$ is connected in $S$ then the image $f(C)$ must also be connected in $T$.
• We then looked at a very important type of continuity known as uniform continuity on the Uniform Continuity of Functions on Metric Spaces page. We said that if $(S, d_S)$ and $(T, d_T)$ are metric spaces with $A \subseteq S$ then a function $f : A \to T$ is said to be Uniformly Continuous on $A$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that for all $x, y \in A$, if $d_S(x, y) < \delta$ then:
(2)
\begin{align} \quad d_T(f(x), f(y)) < \epsilon \end{align}