Sets of Points in One, Two, and Three Dimensions
Before we look at limits of functions containing more than one variable, we will first look at some important definitions regarding sets of points. $\mathbb{R}^n$.
| Definition: Let $P(x_1, x_2, ..., x_n), Q(y_1, y_2, ..., y_n) \in \mathbb{R}^n$. Then for $r > 0$ the $r$-Neighbourhood of $P$ denoted $B_r(P) = \{ Q \in \mathbb{R}^n : d(P, Q) < r \}$ is the set of all points $Q \in \mathbb{R}^n$ such that the distance between $P$ and $Q$, $d(P, Q)$, is less than $r$. |
For example, in $\mathbb{R}$, if $c \in \mathbb{R}$ then $B_r(c) = \{ x \in \mathbb{R} : \mid x - c \mid < r \}$ since $\mid x - c \mid$ represents the distance between $x$ and $c$. In fact, $B_r(c)$ can be rewritten as the open interval centered at $c$ and with radius $r$ denoted $(c - r, c + r)$.
In $\mathbb{R}^2$, if $P(x_1,y_1) \in \mathbb{R}^2$ and $r > 0$ then $B_r(P) = \left \{ Q(x_2, y_2) \in \mathbb{R}^2 : \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} < r \right \}$. Alternatively we can view this set as the set of $Q(x_2, y_2) \in \mathbb{R}^2$ such that $(x_2 - x_1)^2 + (y_2 - y_1)^2 < r^2$ and it should become apparent that $B_r(P)$ represents an open disk centered at $P$ and with radius $r$.
Furthermore, in $\mathbb{R}^3$, if $P(x_1, y_1, z_1) \in \mathbb{R}^3$ and $r > 0$ then $B_r(P) = \left \{ Q(x_2, y_2, z_2) \in \mathbb{R}^3 : \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} < r \right \}$, and we can view this set as the of $Q(x_2, y_2, z_2) \in \mathbb{R}^3$ such that $(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 < r^2$, and once again it becomes apparent that $B_r(P)$ represents an open ball centered at $P$ and with radius $r$.
Of course, we could extend the definition of $r$-neighbourhoods to apply to $\mathbb{R}^n$ in a similar fashion.
| Definition: Let $S$ be a set containing points in $\mathbb{R}^n$. Then the set $S$ is said to be Open in $\mathbb{R}^n$ if $\forall P \in S$ $\exists r > 0$ such that $B_r(P) \subseteq S$, that is for all points $P$ in $S$ there exists an $r$-neighbourhood of $P$ that is contained in $S$. |
For example, the set $S = \{ x \in \mathbb{R} : x > 1 \}$ is open as we can rewrite this set as the open interval $(1, \infty)$. To show this using the definition from above, if $y \in S$ then let $r_y= y - \frac{1 + y}{2} > 0 \}$ and so $B_{r_y}(y) = \{ x \in \mathbb{R} : \mid x - y \mid < r_y$ and so $B_{r_y} (y)$ is the set of $x \in \mathbb{R}$ such that:
(1)Since for all $y \in S$, $y > 1$ and so $\frac{1 + y}{2} > 1$. Therefore $\forall y \in S$ $\exists r > 0$ such that $B_{r_y} (y) \subseteq S$, and so $S$ is open.
Definition: If $S$ is a set in $\mathbb{R}^n$ then the Complement of $S$ denoted $S^c = \{ P \in \mathbb{R}^n : P \not \in S \}$, that is $S^c$ is the set of all points in $\mathbb{R}^n$ that are not contained in $S$.
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One way to think of the complement of a set is like that of a cookie cutter. If a cookie cutter is pressed into cookie dough and the cut piece is removed, then the remaining dough is the "complement" to the cookie. Of course this is a rather silly analogy, however it certain helps in memorizing the definition.
Definition: A point $P$ is a Boundary Point of the set $S$ if $\forall r > 0$, if $\exists Q \in S$ and $R \in S^c$ such that $Q, R \in B_r(P)$, that is every $r$-neighbourhood of $P$ contains points from both $S$ and $S^c$. The set of all boundary points of $S$ is denoted $\mathrm{bdry} (S)$. If $P \in S \setminus \mathrm{bdry} (S)$ then $P$ is said to be an Interior Point of $S$, and if $P \in S^c \setminus \mathrm{bdry} (S)$ then $P$ is said to be an Exterior Point of $S$. The set of all interior points of $S$ is denoted $\mathrm{int} (S)$ and the set of all exterior points of $S$ is denoted $\mathrm{ext} (S)$.
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We note that if $P \in \mathbb{R}^n$ and if $S$ is a set in $\mathbb{R}^n$ then $P$ is either a boundary point, an interior point, or an exterior point.
Now that we have all of these definitions on the back burner, we will now look at the concept of sets of points describing surfaces.