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:

(1)

\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 $d$:

(2)

\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:

(3)

\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:

(4)

\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:

(5)

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