Duran, P.

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NASA TM X-1434 I 
METRIZABILITY AND PATTERN RECOGNITION 
By Peter Duran 
Electronic s Research Center  
Cambridge, Mass.  
NATIONAL AERONAUT ICs AND SPACE ADMINISTRATION 
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https://ntrs.nasa.gov/search.jsp?R=19670029461 2020-03-24T01:15:59+00:00Z
METRIZABILITY AND PATTERN RECOGNITION 
By Peter Duran 
Electronics Research Center 
SUMMARY 
Some attempts have been made to apply metric space theory to pattern recogni- 
tion. The relationship of metrizability to pattern recognition problems will be in- 
vestigated. The number of elements (cardinality) in the set of patterns a lso has  a 
s t rong bearing on the ability to use metr ic  concepts in classification of the patterns. 
The relationship of cardinality to metrizability is discussed. 
INTRODUCTION 
One of the major tasks in pattern recognition is the classification of those 
things which are equivalent with respect to a given criterion. I t  is sometimes con- 
venient to introduce a function on the set of data points which will,  in  some sense, 
indicate how the data points appear to  cluster. Such a function would resemble the 
ordinary distance function that is used in  metric space theory. 
would be partitioned into disjoint, non-empty classes which would exhaust the set i f  
some appropriate equivalence relation induced by the distance function were  used. 
The s e t  of data points 
If a distance function is associated with the se t  of data points, the properties 
of the topology resulting from the metric can be used to classify the data points. F o r  
example, we may define the data points x and y to be  equivalent if they lie in the same 
component of the topological space. A component of a topological space is a maximal 
connected subset of the space. In other words, with respect to  the set of data points, 
a component is a subset of the data points, which is connected and contained in no 
la rger  connected subset of the data points. In any proposed equivalence relation on 
the set  of da ta  points it would be required that the l imit  of a sequence of points of a 
given equivalence class should belong to that class. Since the components of the 
topological space a r e  closed, i t  is appropriate to use them as equivalence classes.  
The s t ruc ture  of a topology determined by a met r ic  is also a function of the 
cardinality of the space. The cardinality of the space will determine in  certain in- 
stances whether or  not the space is metrizable. If the space is metrizable, the 
cardinality of the space will also determine in some instances the s t ructure  of the 
components. 
In some cases i t  will be seen that the components of the data set are just  the 
inriiviiliid cktz p d ~ t s .  This menns that no two data points are equivalent. Under 
other  circumstances it will be  seen that the whole data set is a component. .This i r n -  
plies that all of the data points are equivalent. The following theorems describe the 
usefulness of metrizability concepts in  pattern recognition. A further investigation 
into met r ics  yielding non-trivial partitions of a data set might profitably be pursued. 
ZERO DIRiLENSIONALITY, METRIZABILITY, 
AND THEIR RE LATION TO CARDINALITY 
Many definitions of dimensionality have been given for  very general topological 
spaces. A satisfactory theory of dimension has  been given f o r  separable metric spaces. 
A defining condition for  a topology will be given such that the dimension and metriza- 
bility of the topological space is a function of the cardinality of the space. 
class of spaces will be shown to  be non-metrizable. 
A la rge  
Let X be  a se t  such that 1 < C(X) < N 0, where C(X) is the cardinality of X. 
Since a topology for X i s  TI if, and only if, V x c X, {x} i s  closed, take as a subbase 
for  a topology on X the family of subsets { X - { y} I y c X }. This subbase generates 
the smallest, with respect to  inclusion, TI-topology for X. This topology for X is T2 
and discrete,  since finite and TI  i f ,  and only if, finite and T2 i f ,  and only i f ,  finite and 
discrete.  It is easily shown that this minimal TI-topology fo r  X is compact, separable 
totally disconnected, and satisfies the first and second axioms of countability. 
topology is metrizable by the following metric:  
i f  x # y, then d(x, y) = 1. 
The 
x, y c X, if x = y, then d(x, y) = 0; 
Definitions: A separable metric space X has  dimension zero at a point x F X if, 
and only i f ,  for all neighborhoods of x, N(x), there  is a neighborhood of x, N'(x), such 
that N'(x)E N(x) and the boundary of N'(x) = A separable metr ic  space has dimen- 
sion zero if, and only if ,  it has dimension zero at  each of its points. 
Remark: A compact, separable metric space i s  zero dimensional if ,  and only i f ,  
it is totally disconnected. 
THEOREM 
If X is a non-void finite set with the minimal T1-topologyY then i t  has dimension 
zero. 
PROOF: 
The metrizability and separability of X were indicated above. The assertion 
follows by virtue of the compactness and total disconnectedness of X 
from the fact that V x f X, { x } is open and closed.1 
I t  also follows 
2 
Let X be a set  such that C(X) 2 H 0 .  Again, for  a topology, take as a subbase 
the family of subsets of X, 
imal T1-topology; the family of all finite intersections of the members  of the subbase 
is a base. 
{ X - { y} I y € X } . This subbase generates the min- 
THEOREM 
The minimal T1-topology for  a set X where C(X) = N o  satisfies the first and 
second axioms of countability. 
PROOF: 
The finite intersections of the members  of the subbase correspond to  the finite 
subsets of X, and conversely. 
countably infinite set is Ho.l 
The cardinality of the family of finite subsets of a 
LEMMA 
Each non-empty open se t  in the minimal T1-topology for a set X where C(X) 2 H 
contains all but a finite number of the points of X. 
PROOF: 
Each member of the base is the intersection of a finite number of subbase mem- 
bers ;  hence, i t  contains all but a finite number of the points of X. An arbi t rary union 
of members  of the base contains each subbase member of the union, and therefore,  
contains all but a finite number of the points of X.1 
THEOREM 
The minimal T1-topology for a set X where C(X) 2 Ho is connected and not T2. 
PROOF: 
Assume that X = 0 1  u 02 where 0 1  and 02 a re  non-void open subsets of X and 
infinite number of the points of X. Assume that x ,y  
open subsets of X containing, respectively, x and y. 
would not contain an infinite number of the points of X.1 
o l  n 02 = cp. x - 0 1  = 02 and - ij2 = G1. ii, f"lloii;a 0 ,  az=:! n- nnntain ~ 1 1  hut. an " Z  ----I--- 
X and x # y. 
If O x r \  Oy = c p ,  each of Ox and Oy 
Let 0, and Oy be 
3 
COROLLARY 
The minimal T 1-topology for a set X where C(X) 2 KO is not zero dimensional 
and is not pseudometrizable. 
PROOF: 
The first assertion follows from the fact that the only subsets of X which are 
both open and closed are X and 0. A space is metrizable if ,  and only if, it is T I  and 
pseudometrizable. X is T1 and is not metrizable, since it is not T2; therefore, it i s  
not ps eudome t ri zabl eJ 
THEOREM* 
The minimal TI-topology for a se t  X where C(X) 2 H o  i s  compact. 
PROOF: 
Let (0,) be an open covering of X which is infinite. Select a non-void 0 1  from 
For x € 
1 ,2}  i s  an open 
{ 0, } . 
X - 01, select a 02 in 
covering of X - (01 U 02). 
01, 0 2 , .  . . , On of (0,) which covers X.1 
X - 01  is finite and {Oa I a f 1) is an open covering of X - 01. 
(0, I a # 1)  which contains x. { 0, I a 
Repetition of this process yields a finite subfamily 
LEMMA 
If (X,d) is a metric space such that 2 S C(X) < c y  where c is the cardinality of 
the continuum, then (X, d )  is totally disconnected. 
PROOF: 
Assume that A c X, C(A) 2 2, and A is connected. Define a function f by f(x) = 
d(x,a) where a is a fixed member of A and x 
as the continuous image of a connected set. f(a) = 0. 
> 0;  hence, f(A) is an interval. 
A. f is continuous. f(A) is connected, 
Let b f A and a # b, then f(b) 
It follows that C(f(A)) = c and hence C(A) 2 c.1 
*Theorem is actually a corollary of the lemma on page 3. 
4 
COROLLARY 
' 1  
Let { (Xidi) I i c I }  be an indexed set of metric spaces such that 2 
and V i  c I, 1 s C(Xi) 5 f( 0. Then n (51 i € I} with any metric is totally discon- 
nected. 
C(I) < No 
PROOF: 
Let (X,d) be a pseudo-metric space such that C(X) < c. Define an equivalence 
relation - on X by Vx,  y c X ,  x - y i f ,  and only if ,  d (x,y) = 0. Denote the family of 
equivalence classes  of - by XI and denote by x' the equivalence class  to which x Q X 
belongs. Define a function d' on the Cartesian product of X' with itself by d' (x ' ,yl)  = 
d(a,b)  where a Q x' and b Q y'. (X',d') is a metric space. 
COROLLARY 
(XI, d') is totally disconnected. 
PROOF: 
C(X') 5 c (X). I 
N ON -MET RI Z ABIL ITY THE OR EM 
Every topological space the cardinality of which is less than that of the continuum 
and which contains a connected subset of two or  more distinct points is not metrizable. 
PROOF: 
If  the space w e r e  metrizable, then it would be totally disconnected.1 
COROLLARY 
Every TI-topological space whose cardinality is less than that of the continuum 
and which contains a connected subset of two or more distinct points is non-pseudo- 
metrizabie. 
NASA Electronics Research Center, Cambridge, Massachusetts, August 1967, 
125-03-02-27. 
NASA-Langley, 1967 - 19 c- 16 5 


Metrizability and pattern recognition

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