Open and Closed Sets of Metric Spaces Review

# Open and Closed Sets of Metric Spaces Review

We will now review some of the important information presented recently regarding open and closed sets of metric spaces.

Let $(M, d)$ be a metric space.

- Recall from the
**Open and Closed Balls in Metric Spaces**page that for $a \in M$, the**Open Ball Centered at $a$ with Radius $r > 0$**denoted $B(a, r)$ is defined to be the set of all points in $M$ with distance less than $r$ from $a$ with respect to the metric $d$, that is:

\begin{align} \quad B(a, r) = \{ x \in M : d(x, a) < r \} \end{align}

- Similarly we defined the
**Closed Ball Centered at $a$ with Radius $r > 0$**denoted $\bar{B}(a, r)$ to be defined as the set of all points in $M$ with distance less than*or equal to*$r$ from $a$ with respect to the metric $d$:

\begin{align} \quad \bar{B}(a, r) = \{ x \in M : d(x, a) \leq r \} \end{align}

- On the
**Open Balls in ℝ with the Chebyshev Metric**page we noted that the terminology "open balls" is not necessarily indicative of any actual geometry of these sets. We showed that with the Chebyshev metric in $\mathbb{R}^2$ that the open balls are actually open filled squares.

- With the definition of open balls in a metric space, we then defined some special types of points of sets in a metric space on the
**Interior and Boundary Points of a Set in a Metric Space**page. If $S \subseteq M$ we said that a point $a \in S$ is an**Interior Point**of $S$ if there exists an open ball centered at $a$ that is fully contained in $S$, i.e., $a \in S$ is an interior point of $S$ if there exists a positive real number $r > 0$ such that:

\begin{align} \quad B(a, r) \subseteq S \end{align}

- We said that the set of all interior points of $S$ is called the
**Interior**of $S$ and is denoted $\mathrm{int} (S)$.

- Furthermore, we said that a point $a \in M$ is a
**Boundary Point**of $S$ if every open ball centered at $a$ contains points in $S$ and points in $S^c$. Equivalently, for all $r > 0$ we have that $B(a, r) \cap S \neq \emptyset$ and $B(a, r) \cap S^c \neq \emptyset$.

- We said that the set of all boundary points of $S$ is called the
**Boundary**of $S$ and is denoted $\mathrm{bdry}(S)$.

- On the
**Open and Closed Sets in Metric Spaces**we defined the notion of open and closed sets in a metric space. We said a set $S \subseteq M$ is an**Open Set**if $S = \mathrm{int}(S)$, i.e., every point in $S$ is an interior point of $S$. Furthermore, we said that $S$ is a**Closed Set**if $S^c = M \setminus S$ is open. We also noted that a set may be open AND closed (or neither), and that sets that are both open and closed are said to be**Clopen Sets**.

- We examined the openness/closedness of sets in a particularly abstract example on the
**Open and Closed Sets in the Discrete Metric Space**. We saw that if $d$ is the discrete metric, then every set $S \subseteq M$ is both an open set and a closed set, i.e., every $S$ is a clopen set!

- Furthermore, on
**The Openness of Open Balls and Closedness of Closed Balls in a Metric Space**page we showed if $a \in M$ then every open ball $B(a, r)$ is an open set, and every closed ball $\bar{B}(a, r)$ is a closed set.

- On
**The Union of an Arbitrary Collection of Open Sets and The Intersection of a Finite Collection of Open Sets**page we looked at some very nice theorems. We saw that if $\{ A_i \}_{i \in I}$ is an ARBITRARY collection of OPEN sets then the union $\displaystyle{\bigcup_{i \in I} A_i}$ is also an open set.

- We also saw that if $A_1, A_2, ..., A_n$ is a FINITE collection of OPEN sets then the intersection $\displaystyle{\bigcap_{i=1}^{n} A_i}$ is also an open set.

- Analogously, on
**The Union of a Finite Collection of Closed Sets and The Intersection of an Arbitrary Collection of Closed Sets**page we saw that if $\{ A_i \}_{i \in I}$ is an ARBITRARY collection of CLOSED sets then the intersection $\displaystyle{\bigcap_{i \in I} A_i}$ is also a closed set.

- We also saw that if $A_1, A_2, ..., A_n$ is a FINITE collection of CLOSED sets then the union $\displaystyle{\bigcup_{i=1}^{n} A_i}$ is also a closed set.

- We then proved that every finite set is closed on the
**The Closedness of Finite Sets in a Metric Space**page. We accomplished this by showing that every singleton set $\{ x \}$ for $x \in M$ is closed and then took any finite set and expressed it as a union of a finite collection of singleton sets - hence showing every finite set is closed.

- On the
**Open and Closed Set Differences in Metric Spaces**page we looked at two important theorems regarding the set differences of open and closed sets. We saw that if $A$ is an open set and $B$ is a closed set then the difference $A \setminus B$ is an open set, and the $B \setminus A$ is a closed set.

- On the
**Criterion for a Set to be Open in a Metric Subspace**page we looked into open sets of metric subspaces. We saw that if $(S, d)$ is a metric subspace of $(M, d)$ then a set $X \subseteq S$ is OPEN in $S$ if and only if there exists an OPEN set $A$ in $M$ such that:

\begin{align} \quad X = A \cap S \end{align}

- Similarly, on the
**Criterion for a Set to be Closed in a Metric Subspace**page we looked into closed sets of metric subspaces. We saw that if $(S, d)$ is a metric subspace of $(M, d)$ then a set $Y \subseteq S$ is CLOSED in $S$ if and only if there exists a CLOSED set $B$ in $M$ such that:

\begin{align} \quad Y = B \cap S \end{align}