Difference between revisions of "Metric space"

A set $S$ is said to be a metric space if a real number $d$ may be associated to any two points $p_1$ and $p_2$ in $S$, called the distance from $p_1$ to $p_2$.[1] The most obvious example of a metric space is one-dimensional Euclidean space, where the distance function $d(x,y)$ is simply the absolute value of the difference between $x$ and $y$. In other words, $\forall x,y \in \mathbb{R}$, $d(x,y) = | x - y |$. Metric spaces can be equipped with a natural topology whose basic open sets are the balls $B_{\epsilon}(x)$, where $B_{\epsilon}(x) = \{ y \in X \colon d(x,y)<\epsilon\}$.

Formal Definition

A metric space is an ordered pair $(X,d)$ where $X$ is some set and $d$ is a function $d \colon X \times X \rightarrow [0,\infty)$, called a metric, with the following properties:

1. For all $x,y \in X$, $d(x,y)=d(y,x)$.
2. For all $x,y \in X$, $d(x,y)=0$ if and only if $x=y$.
3. For all $x,y,z \in X$, $d(x,y) \leq d(x,z)+d(z,y)$.

It should be noted that every subset of $S$ is also a metric space in its own right, with the same distance function. Intuitively, it follows that every subset of a Euclidean space is a metric space, and the distance between two points is measured in the same way.

Role in Determining Continuity

The distance metric has implications that go much deeper than may be initially obvious. Using the distance function, the neighborhood of a point may be determined, and subsequently used to find the limit point. With this information, the closure of a set may then be determined.[2] These components are central to determining the continuity of a function, and therefore a topological space.

References

1. Rudin, Walter, and Tata McGraw-Hill Publishing Company. Principles of Mathematical Analysis. Chennai: McGraw Education (india) Private Limited, 2017.
2. Munkres, James R. Topology, 2015.