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Open sets

In the lesson Introduction to set theory you learned about set notation and various properties of sets. In addition, in this lesson we will talk about some specific sets that are subsets of a certain metric space.

Open ball

Definition: Let $x \in \mathbf{R^{n}}$ and $r>0$. A set

$$K(x, r) = \{ y \in \mathbf{R^{n}}: d(x,y)<r\} = \left \{y \in \mathbf{R^{n}} : \sqrt{\sum_{i=1}^{n}(x_{i} – y_{i})^{2}}< r \right \}$$

is called an open ball with center x and radius r.

Note:Expression $d(x, y)$ represents the distance function. In other words, d is the standard Euclidean distance between two points.

Pictures below represent open balls in sets $\mathbf{R}, \mathbf{R^{2}}$ and $\mathbf{R^{3}}$, respectively.




Open sets

Definition: We say that a set $A \subseteq \mathbf{R^{n}}$ is an open set if

$$(\forall x \in A)  \ (\exists r>0)  \ K(x, r) \subseteq A.$$


In other words, every open set is the union of open balls.

You can think of it as a collection of elements that doesn’t include any limit points.

Example 1: An open ball $K(x, r)$ is an open set.

Example 2: An open interval $\left<0, 1\right>$ is an open set in $\mathbf{R}$, but not in $\mathbf{R^{2}}$. More precisely, it is not an open set in $\mathbf{R^{2}}$ because we identify it with a set $\{(x, 0): 0< x < 1 \subset \mathbf{R^{2}}\}$.

Moreover, any open interval is an open set.


Example 3: $\emptyset$ and $\mathbf{R^{n}}$ are open sets.


Topology on a set

Theorem: Let X be a metric space. Furthermore, let $\mathcal{T}$ be a family of all open sets with properties:

(1) $\emptyset, X \in \mathcal{T}$

(2) the union of any family from $\mathcal{T}$ is from $\mathcal{T}$, i.e.

$$U_{\alpha}\in \mathcal{T} \rightarrow \bigcup U_{\alpha}\in \mathcal{T}$$

(3) the intersection of any finite family of elements from $\mathcal{T}$ is an element from $\mathcal{T}$, i.e.

$$U_{i} \in \mathcal{T}, \ i = 1, \cdots, m, \rightarrow \bigcap_{i=1}^{m} U_{i} \in \mathcal{T}.$$

Definition: We say that $\mathcal{T}$ is a topological structure or simply a topology on set X. Furthermore, ordered pair $(X, \mathcal{T})$ is called a topological space.

Elements of a set X are points and elements of a family $\mathcal{T}$ are open sets of the topological space $(X, \mathcal{T})$.


Example 4:  Let $X = \{1, 2, 3\}$ and $\mathcal{T} = \{\emptyset, X, \{1, 2\}, \{1, 3\}\}$. Is $\mathcal{T}$ a topology on X?

Solution: $\mathcal{T}$ is not a topology on because $\{1, 2\} \cap \{1, 3\} = \{1\} \notin X$.


Interior of a set

Definition: Let $A \subseteq \mathbf{R}$. A point $x \in A$ is the interior point of a set A if there exists an open set U such that $x \in U \subseteq A$.

The interior of a set A is a set of all interior points. In other words,

$$IntA = \{x \in \mathbf{R^{n}}: \exists r >0, \ K(x, r) \subseteq A\}.$$

Example 5: Int A is an open set. Moreover, it is the largest open set in A.

Example 6: Interior of a set $S = \{(x, y) \in \mathbf{R}^{2} : 0< x \leq 1\}$ is $Int S = \{(x, y) \in \mathbf{R}^{2} : 0 < x < 1\}$.

Namely, we can put all points $(x, y)$ (for which $0 < x < 1$ is valid) in an open set in S, so all that points are in Int S. On the other side, for a point $(1, y)$ every open set of radius r contains a point which is not an element of S. For instance, point $\left (1 +\frac{r}{2}, y \right)$. Therefore, every open set which contains a point $(1, y)$, also contains a point which is not in S.