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572

List of special symbols

For every symbol of this list we refer the reader to a place where it was defined.There could be many such places, but we only mention one here. Note that a symbolis often defined in the first volume of this book entitled “Topology and FunctionSpaces”; we denote it by TFS. We never use page numbers; instead, we have thefollowing types of references:

(a) To an introductory part of a sectionFor example,

expX 1.1says that expX is defined in the Introductory Part of Section 1.1.Of course,

Cp.X/ TFS-1.1says that Cp.X/ is defined in the Introductory Part of Section 1.1 of the bookTFS.

(b) To a problemFor example,

Cu.X/ TFS-084says that the expression Cu.X/ is defined in Problem 084 of the book TFS.

(c) To a solutionFor example,

O.f;K; "/ S.321says that the definition of O.f;K; "/ can be found in the Solution of Problem321 of the book TFS.The expression,

HFD T.040says that the definition ofHFD can be found in the Solution of Problem 040 ofthis volume.

V.V. Tkachuk, A Cp-Theory Problem Book: Special Features of Function Spaces,Problem Books in Mathematics, DOI 10.1007/978-3-319-04747-8,© Springer International Publishing Switzerland 2014

573

Every problem is short, so it won’t be difficult to find a reference in it. Anintroductory part is never longer than two pages so, hopefully, it is not hard to find areference in it either. Please keep in mind that a solution of a problem can be prettylong, but its definitions are always given in the beginning.

The symbols are arranged in alphabetical order; this makes it easy to find theexpressions B.x; r/ and ˇX , but it is not immediate what to do if we are lookingfor

Lt2T Xt . I hope that the placement of the expressions which start with Greek

letters or mathematical symbols is intuitive enough to be of help to the reader. Evenif it is not, then there are only three pages to plough through. The alphabetic order isby line and not by column. For example, the first three lines contain symbols whichstart with “A” or something similar and lines 3–5 are for the expressions beginningwith “B”, “ˇ” or “B”.

A.�/ TFS-1.2 a.X/ TFS-1.5

AD.X/ TFS-1.4V

A T.300

AjY T.092 Bd .x; r/ TFS-1.3

B.x; r/ TFS-1.3 (B1)–(B2) TFS-006

ˇX TFS-1.3 B.X/ 1.4

clX.A/ TFS-1.1 cl� .A/ TFS-1.1

C.X/ TFS-1.1 C �.X/ TFS-1.1

C.X; Y / TFS-1.1 Cp.X; Y / TFS-1.1

Cu.X/ TFS-084 Cp.Y jX/ TFS-1.5

Cp.X/ TFS-1.1 C �p .X/ TFS-1.1

c.X/ TFS-1.2 conv.A/ 1.2

CH 1.1 �.X/ TFS-1.2

�.A;X/ 1.2 �.x;X/ 1.2

D.�/ TFS-1.2 d.X/ TFS-1.2

dom.f / 1.4 diam.A/ TFS-1.3

D 1.4 �F TFS-1.5

�X TFS-1.2 �.X/ TFS-1.2

�n.Z/ T.019 �ijn .Z/ T.019

} 1.1 }C 1.1

�t2T ft TFS-1.5 ext.X/ TFS-1.2

expX TFS-1.1 Fin.A/ S.326

f <� g 1.1 f � g 1.4

574 List of special symbols

List of special symbols 575

fn!!f TFS-1.1 f jY TFS-022

F� TFS-1.3 F� TFS-1.3

'�.X/ for a card. inv. ' 1.1 F�ı 1.4

Gı TFS-1.3 G� TFS-1.3

Gı� TFS-1.5 GCH 1.1

h#.U / S.228 HFD T.040

h'.X/ for a card. inv. ' 1.1 I TFS-1.1

Int.A/ TFS-1.1 IntX.A/ TFS-1.1

iw.X/ TFS-1.2 J.�/ TFS-1.5

K 1.4 K�ı 1.4

l.X/ TFS-1.2 L.�/ TFS-1.2

limS for S D fAn W n 2 !g 1.5 (LB1)–(LB3) TFS-007

max.f; g/ TFS-1.1 min.f; g/ TFS-1.1

(MS1)–(MS3) TFS-1.3 m.X/ TFS-1.5

MA.�/ 1.1 MA 1.1

nw.X/ TFS-1.2 N TFS-1.1

O.f;K; "/ S.321 O.f; x1; : : : ; xn; "/ TFS-1.2L

t2T Xt TFS-1.4LfXt W t 2 T g TFS-1.4

p.X/ TFS-1.2 (PO1)–(PO3) TFS-1.4

pS W Qt2T Xt !Qt2S Xt TFS-1.4 pt W Qs2T Xs ! Xt TFS-1.2

Qt2T Xt TFS-1.2

QfXt W t 2 T g TFS-1.2Qt2T gt TFS-1.5 ˘0

� .X/ TFS-2.4

.A;X/ TFS-1.2 .x;X/ TFS-1.2

.X/ TFS-1.2 ��.x;X/ TFS-1.4

�w.X/ TFS-1.4 ��.X/ TFS-1.4

�Y W Cp.X/! Cp.Y / TFS-1.5 P 1.1

P 1.4 Q TFS-1.1

q.X/ TFS-1.5 R TFS-1.1

r� for r W X ! Y TFS-1.5 hSi S.489

s.X/ TFS-1.2 SA TFS-2.1

St.x;U/ TFS-1.3 St.A;U/ TFS-1.3

˙�.�/ TFS-1.5 ˙0� .X/ 1.4

�.�/ TFS-1.5 ˙.�/ TFS-1.5

t0.X/ TFS-1.5 tm.X/ TFS-1.5

t.X/ TFS-1.2 �.x;X/ TFS-1.1

�.X/ TFS-1.1 �.A;X/ TFS-1.1

��.X/ TFS-1.2 �.d/ TFS-1.3

�X TFS-1.5Sffi W i 2 I g 1.4

!<! 1.4 U1 ^ : : : ^ Un S.144

!� TFS-370 w.X/ TFS-1.1

Œx1; : : : ; xnIO1; : : : ; On TFS-1.1 X1 � : : : �Xn TFS-1.2

XT TFS-1.2 .Z/! S.493

.Z/� S.493

576 List of special symbols

Index

Aabsolute Borel set of class �, 1.4, 323, 324,

373–375almost closed set, 1.3almost disjoint family of sets, 1.5, 053analytic space, 1.4, 334–340, 352, 353, 360,

361, 363, 366, 368–371, 385–387, 392,395, 397, 399, 460, T.132, T.341, T.363

antichain, 1.1Aronszajn tree, 1.1, 068Aronszajn coding, 1.1, 068

BBaire property, 263–265, 406, 441, T.046,

T.371Baturov’s theorem, 269Booth lemma, 052Borel set, 1.4, 322, 330–334, 339–342, 354,

368, 372, T.322, T.333, T.339, T.372

Ccaliber, 1.3, 275–294, 299, T.281, T.285Cantor set (see also space K), 348, 353, 376,

T.250cardinal function, 1.2, 1.5, 145–151, 405cardinal invariant, 1.1category of a set in a space, 057, T.351, T.371ccc property, 1.1Cech-complete space, 137, 138, 210, 232, 272,

273, 324, 402, 443, T.210, T.272, T.385,T.500

Cech–Stone compactification, 042, 238, 286,371, 386, 387, 403, 439, 491 T.126,T.131, T.244, T.322, T.371, T.385

C -embedded subspace, T.218, T.455, T.468

centered family, T.058, T.280chain, 1.1, 068character at a point, 1.2, 054, T.092character of a set, T.222, T.487, T.489character of a space, 131, 158, 444, T.489clopen set, T.063, T.126, T.203, T.219, T.298,

T.309, T.371closed discrete set, T.126, T.205, T.219, T.285,

T.316, T.455closed cover, 1.3, 373closed map, 206, 245, 316, 478, 493, 494,

T.246, T.493, T.494club (closed unbounded subset of an ordinal),

1.1, 064, 065, T.069cofinal set, 1.1, 097, T.285collectionwise normality, 139, 140, 437 041,

045, 058, 060–062, 072–076, 082–083,085, 090–096, 099, 118–120, 128–131,134, 201–204, 211, 220, 225, 233, 237,241, 248, 254, 259, 260, 262, 273, 279,287, 294–299, 306, 347, 348, 355–357,359, 365, 375, 377, 381, 383–389,391–396, 398, 413, 454, 462, 468,482, 495T.041, T.045, T.062, T.073, T.082,T.090, T.098, T.131, T.211, T.218,T.222, T.227, T.229, T.235, T.250,T.270, T.272, T.298, T.309, T.346,T.357, T.372, T.377, T.384, T.385,T.391, T.395, T.465, T.468, T.487,T.489, T.493

compactification, 1.3, 233, T.224compact-valued map, 1.3, 1.4., 240–242, 249,

388compatible elements of a partially ordered set,

1.1complete family of covers, 1.4

V.V. Tkachuk, A Cp-Theory Problem Book: Special Features of Function Spaces,Problem Books in Mathematics, DOI 10.1007/978-3-319-04747-8,© Springer International Publishing Switzerland 2014

577

Index

completely metrizable space, 315, 316, 373,419, 493, 494, 498, 499, T.132, T.313,T.333, T.348, T.357, T.368

completely regular space, T.139condensation, 045, 077–079, 102, 341,

354–359, 392, T.139, T.250, T.357,T.363

connected space, 1.3, T.309, T.312consistency with ZFC, 1.1, 047continuous map, 094, 104, 105, 121–123, 133,

148, 150, 201, 206, 243, 245–249, 253,254, 277, 305, 315–317, 332, 338, 360,361, 363, 364, 390, 463, 467–473,479–482, 486, 491, T.063, T.069, T.104,T.109, T.131, T.132, T.139, T.237,T.245, T.250, T.252, T.266, T.268,T.294, T.298, T.316, T.318, T.333,T.354, T.368, T.372, T.384, T.459,T.468, T.500

Continuum Hypothesis (CH), 1.1, 039, 040,041, 042, 046, 047, 069, 089, 097-100,237, 238, 298, 300

convex hull, 1.2, 104, T.104convex set, 1.2, T.104cosmic space, 107–111, 192, 195, 198–200,

218, 225, 228, 244, 263, 270, 300, 346,363, 364, 395, 451, T.109, T.250, T.270,T.300, T.363, T.368

countably additive class (or property), 1.5, 405,406, 441, 442, 445–450

countably compact space, 092, 133, 204,218, 226, 417, T.126, T.203, T.235,T.391

countably paracompact space, 141cozero set (see also functionally open set),

T.080, T.252

D�-system, 038�-system lemma, 038�-root, 038dense subset of a partially ordered set, 1.1dense subspace, 009, 010, 039, 040, 072, 080,

237, 239, 278, 368, 385–387, 420, 431,495

dense subspace T.058, T.063, T.074, T.078,T.080, T.081

dense subspace (see also the previous page),T.109, T.187, T.309, T.312, T.333,T.349, T.351, T.368, T.406, T.416,T.421, T.500

dense-in-itself, space, 057, 358, T.045, T.046,T.219, T.272, T.358

density degree, 037, 095, 177, 188, 216, 405,457, 483, 487, T.285

depending on a set of coordinates, map, T.109,T.268, T.298

diagonal of a space, 028, 029, 087, 091, 203,235, 290, 293–298, 300, 396, T.019,T.020, T.021, T.028, T.029, T.030,T.062, T.078, T.081, T.087, T.089,T.098, T.173, T.203, T.294

diagonal number, 028, 029, 091, 178, 179, 180,449, T.087, T.173, T.235

diagonal product of maps, T.266diameter of a set, T.055, T.348, T.358, T.368diamond principle (}), 1.1, 069, 070, 073,

079, 289Dieudonné complete space, 430disconnected space, 1.3, T.219discrete family of sets, T.132, T.217, T.373discrete space, 424, 443, 486, 492 494,

498–500, T.371discrete subspace, 188, 401, T.007, T.098,

T.219, T.500discrete union, T.219, T.250domain of a map, 1.4domination by irrationals, T.391double arrow space, 383Dugundji system, 103, T.104

Eembedding 250, 303, 322, 370, 371, 376–378,

387, T.019, T.132, T.250, T.298, T.333,T.372, T.385

extent, 1.3, 1.5external base, T.092extremal disconnectedness, T.219

Fface of a product, T.109, T.110, T.268, T.298,

T.415, T.455, T.500factorization of a map defined on a subspace of

a product, T.109, T.268faithfully indexed set, T.089filter, T.372filter on a partially ordered set, 1.1finite intersection property (see the entry for

centered family)finitely additive class (or property or cardinal

function), 1.5, 401–403, 406, 407finite-to-one map, 1.5, 498, 499, T.498first category set, 057, T.351first countable space, 099, 202, 329, 401, 408,

415, T.045, T.205, T.351

578

Index

Fodor’s lemma, 067free sequence, 198, T.198free union (see discrete union)Fréchet–Urysihn space, 1.5, 120, 131,

134–136, 186, 204, 205, 210, 214, 384,398, 402, 450, 464–466, T.045

F� -set, 1.4, 323, 379, 420, T.422

GGeneralized Continuum Hypothesis (GCH),

1.1generating topology by a base, 1.3generating topology by a closure operator, 1.3generating topology by a family of local bases,

1.3generating topology by a family of maps, 1.3generating topology by a linear order, 072generating topology by a metric 419generating topology by a subbase, 1.3, T.363generating topology by an interior operator, 1.3Gerlits property, 1.5, 463, 464Gerlits–Pytkeev theorem, 465Gı-diagonal (or diagonal Gı), 028, 029, 087,

235, 300, T.087, T.235Gı-set, 055, 091, T.041, T.062, T.090, T.333,

T.500, T.429G�-set, 1.2, 001, T.078

HHausdorff space, T.098, T.372hedgehog space, 019, 020, 021height of a tree, 1.1height of an element of a tree, 1.1hereditarily analytic space, 400hereditarily Cech-complete space, 272, 273hereditarily irresolvable space, T.219hereditarilyK-analytic space, 400hereditarily k-space, 214hereditarily Lindelöf space, 001, 005, 007,

010, 011, 014, 015, 039, 074, 076, 085,086, 099, 100, 190, 198, T.073, T.270

hereditarily normal space, 002, 090, 142, 201,202, T.090

hereditarily p-space, 272hereditarily realcompact space, 404, 425, 451hereditarily separable space, 1.1, 004, 008,

012, 014, 040, 060, 073, 077–079, 082,088, 089, 098, 198, 237, T.040, T.082,T.099

hereditarily sequential space, 214hereditarily stable space, 200

hereditarily weakly Whyburn space, 215, 220hereditary cardinal function, 146, 147hereditary density, 004, 008, 012, 014, 017,

018, 020, 021, 024, 030, 032, 036, 039,040, 043, 059, 169, 171, 172, 173, 174,405, 428, 458, T.020, T.029, T.030,T.036, T.166, T.173

hereditary Lindelöf number, 001, 005, 007,011, 014, 015, 017, 018, 020, 021, 023,029, 033, 035–037, 039, 040, 043, 059,166, 167–171, 190, 193, 194, 405, 429,458, T.021, T.029, T.030, T.036, T.166,T.173, T.368, T.490

hereditary property, 420, 421, 433, T.455Hewitt realcompactification, 145HFD space, T.040homeomorphic spaces, 313, 330, 347–353,

386, 424, 432, 500, T.132, T.217, T.219,T.250, T.313, T.322, T.348, T.351,T.363, T.415, T.457

homeomorphism, T.217, T.333, T.349homeomorphism (see also previous column),

T.357, T.371, T.372, T.421, T.498,T.500

homogeneous space, 1.5, T.371Hurewicz space, 1.2, 1.3, 132, 188, 216Hurewicz space (see also the previous page),

217, T.132, T.188

Iidentity map, T.357induced topology, 072, T.357invariance under operation, 1.3, 254, T.250irrationals, 4, 313, 317, 328, 329, 341, 347,

352, 358, 359, 365, 367, 370, 371, 388,390, T.132, T.346, T.395

irreducible map, 492, T.246, T.493, T.494irresolvable space, T.219isomorphism of linearly ordered sets, 072i -weight, 177, 178, 244, 425, 451, 471

JJensen’s axiom (see diamond principle (}))

KK-analytic space, 1.4, 343–346, 388, 389, 390,

391, 393–395, 397–400, 460, T.250,T.346, T.391

�-Aronszajn tree, 1.1�-modification of a space, 1.2, 128

579

Index

�-monolithic space, 1.2, 113, 114, 116, 117,120–122, 144, 152, 154, 157, 190–192,197, 199, 296, 426, 468, 473

�-scattered space, 1.2, 133, 187, 477�-simple space, 1.2, 127, 129, 130, 157�-small diagonal, 1.3, 290, 293, 298, 300,

T.294, T.298, T.300�-Souslin tree, 1.1, 070, 071, 073, 074, T.073�-stable space, 1.2, 106, 108–112, 118,

123–126, 143, 152–154, 156, 192, 195,200, 266–268, 478, T.112, T.237

K�ı-space, 1.3, 250, 261, 262, 362, 367,T.250, T.262, T.377

k-space, 1.2, 131, 210, 214, 230, 402, 465,466, T.131, T.210

�-tree (for a regular cardinal �), 1.1, 068, 070,071, 074, T.073

Kowalsky hedgehog J.�/, 019–021, T.019

Llarge inductive dimension, 1.4, 308Lavrentieff theorem, T.333left-separated space, 1.1, 004, 007, 009, 037,

T.004, T.078lexicographic order, 1.4limit of a sequence, 054, 379, 389, T.055,

T.131, T.246, T.316, T.384, T.493limit of a family of sets, 1.5limit of a transfinite sequence, 1.3, T.298Lindelöf number (degree), 001, 128, 240, 269,

405, 456 T.490Lindelöf p-space, 1.3, 223, 231, 244–246, 250,

252, 253, 255, 260, 261, 271Lindelöf property (see also Lindelöf space),

1.2Lindelöf ˙-space, 1.3, 223–228, 230, 231,

233, 234, 236, 239, 242, 243, 248, 249,253, 254, 256–259, 261, 263, 265–270,300, 459, T.227, T.237, T.270, T.399

Lindelöf space, 076, 089, 098, 112, 127, 128,129, 135–137, 189, 199, 234, 264, 294,306, 422, 438, 452, 454, T.112, T.217,T.223, T.268

linear homeomorphism, T.217linear map, T.132linear topological space, 1.2, 104, T.104,

T.105, T.131linearly ordered space, 072–076, T.073,

T.075linearly homeomorphic vector spaces,

T.217local base, T.416

locally compact space, 1.3, 098, 132, 434,T.203, T.223, T.357, T.434

locally convex space, 1.2, 104, T.105, T.131locally finite family, 103, 115, T.104, T.244,

T.354locally pseudocompact space, 435lower semicontinuous map, 1.4, 315L-space, 1.1, 039, 059, 074, 099Luzin space, 1.1, 043–046, 063, T.046, T.063

MMartin’s Axiom (MA), 1.1, 047, 048–063, 071,

083, 088, 099, 140, 197–200, 288, 382,395 T.050, T.063, T.395

maximal almost disjoint family, 053maximalelement of a partially ordered set,

T.074maximal family with a property P , T.058maximal space, T.219measurable map, T.363, T.368, T.384metacompact space, 1.5, 437–440metric space, T.055, T.105, T.313, T.333,

T.348, T.368, T.373metrizable space 062, 083, 090–094, 099,

101–106, 203, 217, 219, 221, 229, 239,272, 285, 295–299, 315, 316, 348–350,357, 359, 373, 374, 392, 395, 396, 401,402, 412–414, 416–419, 446, 451, 455,493–499 T.062, T.132, T.203, T.235,T.285, T.300, T.357, T.385

monolithic space, 1.2, 107, 115–117, 120, 122,152, 154, 155, 266, 297

Mrowka space, 407, T.130multiplicative class of Borel subsets of a space,

1.4multiplicative class of absolute Borel sets, 1.4

Nnatural projection, T.109, T.110, T.250, T.298,

T.455, T.500network, 235network weight, 096, 107–111, 192, 197, 199,

200, 218, 225, 228, 244, 260, 263 270,300, 346, 363, 364, 395, 405, 451, 457,470, 488, T.109, T.250, T.270, T.300

network with respect to a cover, 1.3, T.229normal space, 002, 100, 139–141, 234, 308,

309, 311, 407, 438, 453–455, T.201,T.203, T.217, T.245, T.311, T.372

nowhere dense set, 057, 058, T.039, T.042,T.045, T.089, T.219, T.351

580

Index

O!-cover, 1.5, T.188, T.217, T.464!-monolithic space, 1.2, 116, 117, 120, 122,

190–192, 197, 199, 296, 468, T.468open mapping, 326, 415, 476, 477, 498, 499,

T.110open-separated sets in a topological space,

T.309operator on a family of subsets of a set, 1.3operator on a set, 048, 049, 068, 072, T.004,

T.005, T.074ordinal space, 064, 065–069, 273, 491, T.069,

T.211oscillation of a function, T.368, T.384!-simple space, 1.2, 127!-stable space, 1.2, 106, 112, 119, 125, 126,

145, 192, 195, 267, 268, T.112, T.237outer base, 396

Pparacompact space, 203, 217, 314, 315, 422,

437, 452, T.104, T.203, T.217, T.244,T.245, T.246, T.422

partially ordered set, 048, 049, 068, T.058�-character, 158, 402, 442, T.131, T.158,

T.298P-directed family of sets, 1.4P-dominated space, T.391P.�/-monolithic space for a property P , 1.2,

139, 140, 146–151, 158–186, 189, 190,193, 194, 427–429, 475

perfect image (or preimage), 1.2, 243, 245,249, 252, 304, 325, 390, 487–492,T.245, T.489, T.490, T.492

perfect map, 326, T.266perfect space, 001, T.331, T.363, T.368perfectly disconnected space, T.219perfectly normal space, 003, 061, 075, 080,

081–087, 089, 095, 096, 142, 202,T.080, T.081, T.087

point-countable family of sets, T.203point-finite cellularity, 175, 284, 405, 485,

T.491pointwise bounded subset of function space,

T.384pointwise countable type, T.222Polish space, 1.4, 319, 320, 321, 325, 326–330,

338–340, 347, 351, 358, 365, T.322,T.333, T.339, T.357, T.358, T.384,T.385

P -point, 1.1, 042precaliber, 1.3, 275–280, 283, 284, 286, 288,

289, T.050, T.280

Pressing-Down Lemma, 067product space, 050, 109, 114, 117, 254, 255,

256, 268, 280–282, 302, 333, 335, 343,493, 494–499, T.050, T.089, T.109,T.110, T.112, T.132, T.250, T.266,T.268, T.298, T.363, T.415, T.455,T.500

pseudocompact space, 1.4, 093, 119, 130, 131,138, 205, 435, 452, 495–497, T.132,T.205, T.391, T.465, T.494

pseudocomplete space, 1.5, T.498pseudocharacter of a space, 179, 236, 401, 409,

448pseudometric, 1.4pseudo-open map, 1.3, 251pseudoradial space, 1.3, 211, 212p-space, 1.3, 221, 222–224, 230–232p-space (continued from the previous page),

244–247, 250, 251, 252, 253, 255, 260,271, 272, T.222, T.223, T.224, T.245

P -space, 1.1, 1.2, 112, 127, 135, 137, T.112�-weight, 131, 406, 442, T.089, T.158, T.187

Qquotient image, 149, 251, 274, 275

Rradial space, 1.3, 209, T.211rational numbers, the space of, 349–351, 356,

T.309, T.349, T.351real line, 342, 355, 380, 382, T.349realcompact space, 391, 404, 423, 425, 430,

436, 451, T.436, T.492resolvable space, T.219restriction map, T.080, T.217, T.218, T.368,

T.455, T.500retract, 1.2, 123, 316, 500, T.132, T.217, T.500retraction, 1.2right-separated space (this coincides with the

concept of scattered space), 1.1, 005,006, 008, T.005

Rosenthal compact space, 1.4, 383–385, 387,T.385

R-quotient map (or image), 147, 149, 150, 181,183, 184, 469, T.139, T.268

SSA axiom, 1.1, 036, 086, 193, 195, 196scattered space, 1.1, 006, 098, 099, 128–130,

133, 134, 136, 213, 272, 273, T.130,T.272

581

Index

� -compact space, 132, 216, 226, 274, 323,351, 352, 354, 355, 362, 364, 366, 367,368, 466, 482, T.132, T.203, T.372,T.395

� -countably compact space, 132� -discrete family, 373, T.229, T.235� -discrete network, 228, 235, T.229� -disjoint base, T.412second category set, T.371second countable space, 046, 055, 057, 080,

084, 102, 107, 109–111, 229, 248, 249,252, 254, 271, 306, 318, 320, 321,323, 324, 329, 332, 333, 341, 351,362–365, 367, 368, 376–379, 403, 411,455, T.055, T.062, T.063, T.080, T.089,T.092, T.131, T.132, T.220, T.250,T.298, T.320, T.322, T.341, T.351,T.354, T.363, T.368, T.377, T.379,T.384, T.455

separable space, 039, 044, 046, 073–076, 081,084, 088, 089, 217, 282, 287, 385, 397,412, 493, 495, T.045, T.073, T.074,T.081, T.082, T.087, T.089

separable metrizable space (in this book this isthe same as second countable space),106, 219, 493, 497–499

separating points by a family of maps, T.354separating subsets by a family of sets, 1.3,

233separating subsets by disjoint open (or Borel)

sets, T.309, T.339separation axioms, T.098sequential space, 041, 131, 210, 211, 214, 402,

465, 466, T.041, T.316sequentially compact space, T.384Shanin condition, 282simple space, 1.2, 129, 130� -locally compact space, 132small diagonal (see also �-small diagonal), 1.3,

290, 293, 294, 295, 296–298, T.298,T.300

Sorgenfrey line, 227Souslin line, T.074Souslin number, 075, 275, 405, 485, T.050,

T.089Souslin property, 050, 058, 288, 289, T.039,

T.046, T.050, T.073Souslin tree (see �-Souslin tree)space A.�/, T.203, T.219, T.223space ˇ!, 042, 238, 371, 386, 387, 403, 439,

491, T.042, T.131, T.322, T.385space D.�/ (see discrete space)

space D� (see also Cantor cube), 039, 040,

089, 303, T.040, T.298, T.372space I, 133, 143, 144, 305, T.436, T.500space I

� , 354, 369, T.250, T.298space J.�/ (see Kowalsky hedgehog)space K (see also Cantor set), 1.4, 318, 376,

378, T.250, T.348space L.�/, 227, 402, 440, T.220space P (see irrationals)space Q (see rational numbers)space R (see real line)space R

� , 117, 330, 354, 360, 361, 367, 371,381, 382, 424, 492, 494, 500, T.019,T.132, T.217, T.312, T.372, T.379,T.384, T.385, T.399, T.434, T.455,T.500

space �.A/, 1.2, T.312space ˙.A/, 1.2space !1, 064–069, 491, T.069space !1 C 1, 273, T.211spread, 007, 008, 013, 014, 016, 019, 022, 028,

031, 034, 036, 037, 039, 040, 060–062,078, 079, 160, 161, 163, 164, 189, 190,191, 192, 193, 195, 196, 197, 199, 405,427, 458, T.007, T.015, T.019, T.028,T.036, T.078, T.082, T.087, T.098,T.160

� -product, 1.2, T.110, T.268˙-product, 1.2, T.110, T.268� -pseudocompact space, 132, 435� -space, 228, 235˙-space, 1.3, 221, 223, 224, 226, 228, 229,

234–236, 238, T.223, T.229S-space, 1.1, 040, 059, 098, 099stable space, 1.2, 108–111, 118, 124, 152–154,

156, 200, 268stationary set, 1.1, 065, 066, 067, 069strictly weaker topology, 072strong L-space, 1.1, 059, 099strong S-space, 1.1, 059, 098, 099strong ˙-space, T.223, T.229strongly dense subspace, 1.5strongly �-monolithic space, 1.2, 135, 157, 158strongly monolithic space, 1.2, 101, 136strongly zero-dimensional space, 1.4, 306–312,

314–316, T.306, T.311subbase of a topology, 1.1submaximal space, 1.3, 208, T.219submetrizable space, 1.4, 392subparacompact space, T.223, T.235subtree, 1.1supremum of a family of topologies, T.139

582

Index

TTalagrand space, 234tightness of a space, 120, 176, 183–185, 207,

234, 287, 296, 410, 447, 456, 461, 462,463, T.041, T.173, T.298, T.384

topology of uniform convergence (this is thesame as uniform convergence topology)

tree, 1.1, 068, 070, 071, 074, T.073T0-separating family, T.270T1-separating family, T.203, T.270T1-space, T.219T2-space (see Hausdorff space)T3 12

-space (see Tychonoff space)T4-space (in this book this is the same as

normal space)two arrows space (see double arrow space)Tychonoff cube (see space I

� )Tychonoff space T.098, T.139, T.205, T.219,

T.245, T.363, T.372

Uultradisconnected space, T.219ultrafilter, T.058, T.371uniform convergence topology, 431, T.357,

T.379, T.421uniformly dense set, 1.5, 456–461, T.457,

T.459uniformly discrete set in a metric space,

T.373

upper semicontinuous map, 1.3, 240–242, 249,388, T.346

VVietoris topology, T.372Velichko’s theorem, 030

Wweak functional tightness, 181, 182, 484weaker topology, 072weakly Whyburn space, 1.3, 206, 209, 211,

213, 215, 219, 220, T.220weight, 091, 094, 096, 102, 105, 131, 187,

244, 295, 303, 304, 403, 405, 411,445, T.039, T.046, T.089, T.092, T.102,T.105, T.109, T.158, T.187, T.250,T.268, T.285, T.322, T.357, T.372,T.412, T.487, T.489, T.490

Whyburn space, 1.3, 204, 205–212, 216–219,T.217, T.219

Zzero-set (see also functionally closed set), 1.5,

T.080, T.252zero-dimensional space, 1.1, 034, 035,

301–307, 309, 312, 313, 347, 348,T.063, T.132, T.205, T.298, T.306,T.309, T.311

583