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Metric Space

[] Let X be any unstructured nonempty set.

[] Let d:X×XR be a function that assigns a real number to any pair of elements in the set X. Intuitively, it takes a pair of elements in the set X as the input and provides the distance between them (some real number) as the output. We could also say that d takes an element from the set X×X as the input and produces a real number as the output.

[] The tuple (X, d) is called a metric space if the following three conditions hold.

Nonnegativity: d(x,y)0x, yX, and d(x,y)=0 iff x=y.

Symmetry: d(x,y)=d(y,x)x, yX.

Triangular inequality: d(x,y)d(x,z)+d(z,y)x, y, zX.

[] The function d is called a metric on the set X.

[] Unless necessary, we will denote the metric space as X.

[] εneighborhood of x in X: For any xX and εR++, the εneighborhood of x in X is

the set Nε,X(x)={yX:d(x,y)<ε}.

[] Open subset of X: A subset S of X is said to be open in X if

for each xS,

εR++ such that,


[] Closed subset of X: A subset S of X is said to be closed in X if

XS is open in X.

[] SX may be open, not open, closed, not closed, clopen, or neither open nor closed.

[] Closure of a subset of X: Given any SX, the closure of S in X is

the smallest closed set ClX(S) such that

SClX(S) .