Number Theory: An Introduction to Mathematics

28 I The Expanding Universe of Numbers

In any metric space there is a naturaltopology. A subsetGof a metric spaceE
isopenif for eachx∈Gthere is a positive real numberδ=δ(x)such thatGalso
contains the whole open ballβδ(x)={y∈E:d(x,y)<δ}.AsetF⊆Eisclosedif
its complementE\Fis open.
For any setA⊆E, itsclosureA ̄is the intersection of all closed sets containing it,
and itsinteriorintAis the union of all open sets contained in it.
A subsetFofEisconnectedif it is not contained in the union of two open subsets
ofEwhose intersections withFare disjoint and nonempty. A subsetFofEis (se-
quentially)compactif every sequence of elements ofFhas a subsequence converging
to an element ofF(andlocally compactif this holds for every bounded sequence of
elements ofF).
Amapf:X→Yfrom one metric spaceXto another metric spaceYiscontin-
uousif, for each open subsetGofY,thesetofallx∈Xsuch that f(x)∈Gis an
open subset ofX. The two properties stated at the end of§3 admit far-reaching gen-
eralizations for continuous maps between subsets of metric spaces, namely that under
a continuous map the image of a compact set is again compact, and the image of a
connected set is again connected.
There are many examples of metric spaces:

(i) LetE=Rnbe the set of alln-tuplesa=(α 1 ,...,αn)of real numbers and define

d(b,c)=|b−c|,

whereb−c=(β 1 −γ 1 ,...,βn−γn)ifb=(β 1 ,...,βn)andc=(γ 1 ,...,γn),and

|a|=max

1 ≤j≤n

|αj|.

Alternatively, one can replace thenorm|a|by either

|a| 1 =

∑n

j= 1

|αj|

or

|a| 2 =

(∑n

j= 1

|αj|^2

) 1 / 2

.

In the latter case, d(b,c)istheEuclidean distancebetweenbandc. The triangle in-
equality in this case follows from theCauchy–Schwarz inequality: for any real numbers
βj,γj(j= 1 ,...,n)

(∑n

j= 1

βjγj

) 2

≤

(∑n

j= 1

β^2 j

)(∑n

j= 1

γ^2 j

)

.

(ii) LetE =Fn 2 be the set of alln-tuplesa=(α 1 ,...,αn),whereαj =0or1
for eachj, and define theHamming distanced(b,c) betweenb=(β 1 ,...,βn)and
c=(γ 1 ,...,γn)to be the number ofjsuch thatβj=γj. This metric space plays a
basic role in the theory oferror-correcting codes.