PART IV Appendices SPECIAL REFERENCE CHARTS The next few pages contain six basic reference charts which display the properties of the various examples. The properties of a topological space have been grouped into six nearly disjoint categories: separation, com­ pactness, paracompactness, connectedness, disconnectedness, and metriza­ tion. In each category we have listed those spaces whose behavior is par­ ticularly appropriate. We usually chose any space which represented a counterexample in that category or which exhibited either an unusual or an instructive pathology; occasionally we listed a space simply because it was so well behaved. Entries in the charts are either 1, 0, or -, meaning, respectively, that the space has the property, does not have the property, or that the property is inapplicable. Occasional blanks represent properties which were not dis­ cussed in the text and which do not appear to follow simply from anything that was discussed. Examples are listed by number, and in a few cases the tables extend beyond one page in length. 185 Special Reference Charts 187 Table I SEPARATION AXIOM CHART -<~ ;.. ..:l ..:l ~ p IZ1 ~ ~ Z Eo< -< ~ ~ ~ ~ -< ~ IZ1 ..:l ..:l ;j ::;l ::;l ..:l 0 ..:l P -< t; ....~ p en Po. ~ ::;l Po. = re g;; ., ::;l g;; ::;l 0 g;; 0 ., .. IZ1 ~ 0 OZIZ1Z E-f E-7 E-<" ~- E-< E-<- E-< E-<'" 00 &'!" P 8~ Z u p.. 4 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 17 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 18 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 28 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 52 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 53 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 54 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 55 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 57 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 63 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 66 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 75 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 78 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 79 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 80 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 81 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 82 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 84 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 86 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 88 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 90 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 91 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 92 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 93 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 94 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 100 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 101 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 103 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 105 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 126 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 188 Appendix Table II COMPACTNESS CHART Eo< Q -< 00 '"f>;1 ""::>l Z Eo< 0 Eo< Q U Q Eo< p; p; Q ~ Eo< ..1 Q f>;1 p; ::>l ::>l ..1 Z 1%1 Eo< -< f>;1 0 -< Q 0 1%1 ::>l U Eo< -< U -< ..1 ~ 0 Eo< ::>l Eo< III ~ Z Q ..1 0"" -< U ~ p -< ::>l Z Eo< U ..1 "" -< U ..1 0 0 Q P Z f>;1 Eo< ~ 0 0 f>;1 ..1 -< U ""::>l U f>;1 p ..1 ..1 Eo< III 0 ....:I ..1 U 0 III III -< :0 E:: ~ ::l Q '" -< ..1 III U -< 0;,-< ""..1 Eo< Z 0'" ..1 c:> t:l Eo< p; f>;1 f>;1 ::l t:l ..1 -< -< ... ""~ Z ~ Z Il:: Z Eo< Z Il:: 0 t:l P -< P -< '" -< 0 Eo< ~ P 0' f>;1 f>;1 g P f>;1 0 U Z 0 0 .3 '"Il:: 0 I f>;1 '" Eo< I ""f>;1 '"f>;1 U b ~ U CD. ~ p....'" ....:I CD. b CD. CD. ~ u ~ 2 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 3 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 6 0 1 1 0 0 1 0 1 1 1 1 1 1 1 0 9 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 10 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 15 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 19 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 20 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 21 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 22 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 25 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 26 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 28 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 30 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 31 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 36 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 42 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 43 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 47 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 48 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 50 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 51 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 52 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 57 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 58 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 60 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 63 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 65 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 Special Reference Charts 189 Table II (continued) Eo< 0 -< Po< '"IZ1 ::iii Z Eo< Eo< 0 8 0 Eo< 0 ~ I>- ~ Eo< ;:;! ..;! )1 0 Eo< ~ Z ~ 0 0 0 -< IZ1 )1 =-< Po< ..;! Q Eo< Eo< -< Q :c! 0 Po< )1 =~ ~ Z 0 ..;! 0 -< Q I>- p ;:;! Z =Eo< Q ..;! -< Q p ..;! 0 ~ 0 0 Z IZ1 Eo< Q )1 Q 0 0 P ..;! 0 ~ -< 0 I>- Q ~ Eo< JO. E:: ~ ..;! ~ 0 0 :0 0 I>- ..;! =-< ~ ..;! = Z ~ 0 ..;! Q Q = N -< ~ Q ..;! -< = ~ ... Po< )1 IZ1 ~ Z 0 ~ Z Eo< 0 f::l z p p -< 0" 0 Z g: )1 p CJ -< IZ1 0 go: -< go: p 0 Q Z 0 IZ1 .3 0 '" IZ1 I IZ1 0 IZ1 IZ1 0 Q b J Q 00 ~ p....'" ~ J3 b 00'" 00 ~ Q ~ 66 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 68 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 70 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 71 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 75 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 76 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 77 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 78 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 79 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 82 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 87 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 89 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 98 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 99 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 100 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 101 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 102 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 103 0 0 0 0 0 0 0 0 0 0 0 0 0 105 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 106 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 107 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 109 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 111 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 112 0 0 0 1 1 1 0 1 0 1 0 113 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 132 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 136 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 139 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 140 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 141 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 190 Appendix Table III PARACOMPACTNESS CHART e- e- 0 o ~ e- ~ ::;t 0 ::;t 0 0 0 ~ 0 -< e--< ~ 1>1 e- o ~ e- 0 O ~ ::is 0 ::;t i>< i>< i>< I>: fj ~ ~ 0-< 0-< 0-< 0 e- ::;t III III III ~.