@l $f $$c ,gloss>y $p c4c4c43 ,a topological space has ! $$ub c4c4c4
7c.ta# *a9 3di;n7 $$uf if "! is no collec;n ( unc.tably _m mutu,y 4jo9t
op5 subsets (! space4 $p collec;nwise ,hausdorff3 ,a @m ,t1-space @l ;,x
is $$ub collec;nwise ,hausdorff $$uf if :5"e @m .(x;.a"\ .a @e ,a.) @l is
a 4crete collec;n ( po9ts ( @m ;,x_1 @l "! is a mutu,y 4jo9t collec;n @m
.(,u;.a"\ .a @e ,a.) @l ( op5 subsets ( @m ;,x @l ) @m x;.a @e ,u;.a @l =
e @m .a @e ,a_4 @l $p collec;nwise normal3 ,a topological space ;,x is
$$ub collec;nwise normal $$uf if :5"e @m .(,f;.a"\ .a @e ,a.) @l is a
4crete collec;n ( clos$ subsets ( @m ;,x_1 @l "! is a mutu,y 4jo9t
collec;n @m .(,u;.a"\ .a @e ,a.) @l ( op5 subsets ( ;,x ) @m ,f;.a _"k
,u;.a @l = e @m .a @e ,a_4 @l $p c.tably p>acompact3 ,a topological space
is $$ub c.tably p>acompact $$uf if e c.ta# op5 cov] (! space has a loc,y
f9ite op5 ref9e;t4 $p developa#3 ,a topological space is $$ub developa#
$$uf if x has a develop;t4 @l $p develop;t3 ,a $$ub develop;t $$uf =a
topological space ;,x is a collec;n @m .(@;,g;n"\ @l n @m @e .w.) @l ( op5
cov]s
 
@l $p 4crete3 ,a collec;n @m @;,f @l ( subsets (a topological space ;,x is
$$ub 4crete $$uf if e po9t ( ;,x has a nei<borhood : meets at mo/ "o memb]
( @m @;,f_4 @l ,a collec;n ( po9ts ( ;,x is $$ub 4crete $$uf if !
associat$ collec;n ( s+letons is 4crete4 $p loc,y c.ta#3 ,a collec;n @m
@;,u @l ( subsets (a topological space is $$ub loc,y c.ta# $$uf if e po9t
(! space has a nei<borhood : meets only c.tably _m memb]s ( @m @;,u_4 @l
$p loc,y f9ite3 ,a collec;n @m @;,u @l ( subsets (a topological space is
$$ub loc,y f9ite $$uf if e po9t (! space has a nei<borhood : meets only
f9itely _m memb]s ( @m @;,u_4 @l $p metacompact3 ,a topological space is
$$ub metacompact $$uf if e op5 cov] (! space has a po9t-f9ite op5 ref9e;t4
$p meta-,l9del(3 ,a topological space ;,x is $$ub meta-,l9del( $$uf if e
op5 cov] ( ;,x has a po9t-c.ta# op5 ref9e;t4 $p monotonic,y normal3 ,a @m
,t1-space @l ;,x is $$ub monotonic,y normal $$uf if "! is a func;n ;,h
7"kn z a monot"o normal;y op]ator7 : assigns 6ea* ord]$ pair @m (x, ,u)_1
@l ": @m x @e ,x @l & ;,u is an op5 nei<borhood ( ;x 9 ;,x1 an op5 set @m
,h(x, ,u) @l s* t = ele;ts ;x & ;y ( ;,x & op5 sets ;,u & ;,v 9 ;,x3 @m $l
(i) @l if x @m @e ,u_1 @l !n x @m @e ,h(x, ,u); $l (ii) if x @e ,u _"k
,v_1 @l !n @m h(x, ,u) _"k ,h(x, ,v); @l & @m $l (iii) if x /.k y_1 @l !n
@m ,h(x, ,x-.(y.)) .% ,h(y, ,x-.(x.)) .k .f_4
 
@l $p ,moore space3 ,a @m ,t3-space @l is a $$ub ,moore space $$uf if x is
developa#4 $p p>acompact3 ,a topological space is $$ub p>acompact $$uf if
e op5 cov] (! space has a loc,y f9ite op5 ref9e;t4 $p p>a-,l9del(3 ,a
topological space is $$ub p>a-,l9del( $$uf if e op5 cov] (! space has a
loc,y c.ta# op5 ref9e;t4 $p po9t-c.ta#3 ,a collec;n @m @;,u @l ( subsets
(a topological space is $$ub po9t-c.ta# $$uf if e po9t (! space 2l;gs to
at mo/ c.tably _m memb]s ( @m @;,u_4 @l $p po9t-f9ite3 ,a collec;n @m @;,u
@l ( subsets (a topological space is $$ub po9t-f9ite $$uf if e po9t (!
space 2l;gs to at mo/ f9itely _m memb]s ( @m @;,u_4 @l $p @m .s-_ @l
4jo9t3 ,a collec;n ( subsets (a topological space is @m ".s-disjoint%:] @l
if x is ! union ( c.tably _m mutu,y 4jo9t collec;ns1 ea* ( : is mutu,y
4jo9t4 $p @m .s-_ @l loc,y c.ta#3 ,a collec;n ( subsets (a topological
space is @m ".s-locally%:] @l c.ta# if x is ! union ( c.tably _m loc,y
c.ta# collec;ns4 $p @m .s-_ @l loc,y f9ite3 ,a collec;n ( subsets (a
topological space ;,x is @m ".s-locally finite%:] @l if x is ! union (
c.tably _m loc,y f9ite collec;ns4 $p @m .s-_ @l p>acompact3 ,a topological
space is @m ".s-paracompact%:] @l if e op5 cov] ( ;,x has a @m .s-_ @l
loc,y f9ite op5 ref9e;t4 $p @m .s-_ @l p>a-,l9del(3 ,a topological space
is @m ".s-para,lindelof%:] @l if e op5 cov] (! space has a @m .s-_ @l
loc,y c.ta# op5 ref9e;t4 $p scre5a#3 ,a topological space is $$ub scre5a#
$$uf if e op5 cov] (! space has a @m .s-_ @l 4jo9t op5 ref9e;t4 $p /> ( ;x
9 @m @;,u_3 @l ,giv5 a collec;n @m @;,u @l ( subsets (a topological space
;,x &a po9t @m x @e .+ @;,u_1 @l ! $$ub /> $$uf ( ;x 9 @m @;,u_1 @l denot$
by @m st(x, @;,u)_1 @l is ! union ( all ( ^? memb]s ( @m @;,u @l 6: ;x
2l;gs4 @l $p />-c.ta#3 ,a collec;n @m @;,u @l ( subsets (a topological
space is $$ub />-c.ta# $$uf if ea* memb] ( @m @;,u @l has n"ompty 9t]sec;n
) only c.tably _m memb]s ( @m @;,u_4 @l $p />-f9ite3 ,a collec;n @m @;,u
@l ( subsets (a topological space is $$ub />-f9ite $$uf if ea* memb] ( @m
@;,u @l has n"ompty 9t]sec;n ) only f9itely _m memb]s ( @m @;,u_4 @l $p
/r;gly collec;nwise ,hausdorff3 ,a @m ,t1-space @l ;,x is $$ub /r;gly
collec;nwise ,hausdorff $$uf if :5"e @m .(x;.a"\ .a @e ,a.) @l is a 4crete
collec;n ( po9ts ( @m ;,x_1 @l "! is a 4crete collec;n @m .(,u;.a"\ .a @e
,a.) @l ( op5 subsets ( ;,x ) @m x;.a @e ,u;.a @l = e @m .a @e ,a_4