Jonginn Yun

Metric Spaces

A metric space is a set $X$ equipped with a distance function

$$d:X\times X\to \mathbb R_{\ge 0}$$

satisfying positivity, symmetry, and the triangle inequality.

Metric spaces are the bridge between analysis and topology. They give a concrete source of open balls,

$$B_r(x)=\{y:d(x,y)<r\},$$

and therefore a topology.

Convergence

A sequence $x_n$ converges to $x$ if for every $\varepsilon>0$, there exists $N$ such that

$$n\ge N\Rightarrow d(x_n,x)<\varepsilon.$$

This definition is numerical, but the idea is topological: eventually the sequence enters every neighborhood of the limit.

Compactness

In metric spaces, compactness can often be read through sequences: every sequence has a convergent subsequence. In general topology, compactness is phrased using open covers.

Compactness is useful because it turns local control into finite control. This is why compactness appears repeatedly in geometry, analysis, and physics.

Continuity

A function $f:X\to Y$ between metric spaces is continuous if

$$d_X(x_n,x)\to 0 \Rightarrow d_Y(f(x_n),f(x))\to 0.$$

Equivalently, inverse images of open sets are open. The second formulation is the one that survives after the metric is forgotten.