## David J. Lutzer

## Mathematics Department

## College of William and Mary

August 5, 2015

## Education:

- B.S., Creighton University (1964)
- Diploma in Advanced Mathematics, Oxford University (1966)
- Ph.D., University of Washington (1970)

## Position:

## Research Interests:

- General and set theoretic topology, ordered
spaces, function spaces, domain representability of
topological spaces, domain theory.

## Undergraduate Honors Students

- Cynthia Bowers, 1999, "Linear Orderings: A Sampler"

- Will Funk, 2002, "Branch Spaces and Lexicographic Orderings of Trees"

- Kevin Duke, 2006, "Domain Theory: Modeling Generalized Ordered
Modifications of the Real Line"

- Jacob Hill, 2009, "The Set of All Countable Ordinals: An Inquiry into its
Construction, Properties, and a Proof Concerning Hereditary
Subcompactness."

## Recent Publications:

"Semi-stratifiable spaces with monotonically normal compactifications"
(with H. Bennett), Topology and its Applications, 197(2016), 21-27.
pdf version

"Mary Ellen Rudin and monotone normality" (with H. Bennett), Topology and
its Applications, 195(2015), 50-62. pdf
version

"The Gruenhage property, property *, fragmentability, and sigma-isolated
networks in generalized ordered spaces"
(with H. Bennett), Fundamenta Mathematicae, 223(2013), 273-294. pdf version

"A shorter proof of a theorem on hereditarily
orderable spaces" (with H. Bennett), Topology Proceedings,
40(2012), 311-318.
pdf version

"Choban operators in generalized ordered spaces"
(with H. Bennett and D. Burke), Topology and its Applications, 160(2013), 633-647.
pdf version

"Some Questions of Arhangel'skii on Rotoids" (with
H. Bennett and D. Burke), Fundamenta Mathematicae, 216(2012),147-161.
pdf version

"The Big Bush Machine" (with H. Bennett and D. Burke),
Topology and its Applications, 159(2012), 1514-1528.
pdf version

"Measurements and G-delta subsets of domains",
(with H. Bennett), Canadian
Mathematical
Bulletin, 54(2011), 193-206.
pdf version

"A note on monotonically metacompact spaces" (with
H. Bennett and K.P. Hart), Topology and its
Applications, 157(2010), 456-465.
pdf version

"Strong completeness properties in topology" (with H.
Bennett), Questions and Answers in General
Topology, 27(2009), 107-124. pdf version

"Domain-Representable Spaces and Completeness",
(with H. Bennett),
Topology Proceedings, 34(2009), 223-244.
pdf version

"Subcompactness and Domain Representability in GO-spaces on Real
Numbers" (with H. Bennett),
Topology and its Applications, 156(2009), 939-950. pdf
version

"CBMS2005: Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States" (with S. Rodi, E. Kirkman, and J. Maxwell), American Mathematical Society, Providence RI, 2007, 354 pp. pdf version

"Domain representability of Cp(X)" (with H. Bennett),
Fundamenta Mathematicae, 200(2008), 185-199.
pdf version

"Domain representability of certain function spaces"
(with H. Bennett), Topology and its Appl.,156
(2009), 1937-1942.
pdf version

"Scott representability of some spaces of Tall and
Miskin" (with H. Bennett), Applied General
Topology, 9(2008), 281-292.
pdf version

"Selected ordered space problems'' (with H. Bennett), pp. 3-9
in "Open Problems in Topology, II," ed. by
E. Pearl, Elsevier, New York,2007. pdf version

"Amsterdam properties of Cp(X) imply discreteness of X" (with J. van Mill and
V. Tkachuk), Canadian Math. Bulletin, 51(2008), 570-578.
pdf version

"Scott-domain representability of a class of generalized ordered spaces" (with
K.W. Duke), Topology Proceedings, 32(2008), 1-14.
pdf version

"Domain-representability and the Choquet game in
Moore and BCO spaces" (with H. Bennett and
G.M. Reed), Topology and its Applications,
155(2008), 445-458.
pdf version

"A countably-based domain representation of a
non-regular Hausdorff space" (with H. Bennett), Topology Proceedings, 30(2006),
83-93.
pdf version

"Domain-representability of certain complete
spaces" (with H. Bennett), Houston Journal of
Mathematics, 34(2008), 753-772.
pdf version

"Compact G-delta sets" (with H. Bennett and
R. Byerly), Topology and its Applications, 153(2006), 2169-2181.
pdf version

"The NSF Budget: Not Far Away From Us", FOCUS
25(2005), 32.

"Domain-representable spaces" (with H. Bennett),
Fundamenta Math., 189(2006), 255-268.
pdf version

"On a question of Maarten Maurice" (with
H. Bennett), Topology and its Applications, 153(2006), 1631-1638.
pdf version

"Science Policy at the MAA," FOCUS
24(2004), 10-11.
pdf version

"Quarter-stratifiability in ordered spaces" (with
H. Bennett), Proc. Amer. Math.Soc., 134(2006), 1835-1847. pdf version

"The beta-space property in generalized ordered
and monotonically normal spaces" (with H. Bennett),
Topology and its Applications, 153(2006), 2218-2228. pdf
version

"The arithmetic of algebraic numbers: an
elementary approach" (with C-K Li), College
Mathematics Journal, 35(2004), 307-309.
pdf version

"Lexicographically ordered trees" (with Will
Funk), Topology and its Applications, 151(2005), 275-300.
pdf version

"The monotone Lindelof property and separability
in ordered spaces" (with H. Bennett and
M. Matveev), Topology and its Applications,
151(2005), 180-186.
pdf version

"Branch space representations of lines" (with Will
Funk), Topology and its Applications,
151(2005), 187-214.
pdf version

"Staffing Shifts in Mathematical Sciences
Departments, 1990-2000," (with J. Maxwell),
Notices American Mathematical Society, 50(2003),
683-686.
pdf version

"Mathematics Majors 2002," Notices American
Mathematical Society, 50(2003), 235-237.
pdf version

CBMS2000: Statistical Abstract of Undergraduate
Programs in the Mathematical Sciences in the
United States, (with J. Maxwell and S. Rodi),
American Mathematical Society, Providence, RI,
2002, 303 pages. pdf version

"Spaces with $< \omega$-weakly uniform bases,"
with H. Bennett, Topology Proceedings, 27(2003).
pdf
version

"Metrically fibered generalized ordered spaces,"
with H. Bennett, Topology and its Applications,
130(2003), 283-290.
pdf version

"Lines, trees, and branch spaces," with H. Bennett
and M.E. Rudin, ORDER, 19(2002), 367-384.
pdf version

"Linearly ordered and generalized ordered
spaces," with H. Bennett, pp. 326-330 in
Encyclopedia of General Topology, ed. by
K.P. Hart, J. Nagata, and J. Vaughan, Elsevier,
Amsterdam, 2004.
pdf version

"Continuous separating families in ordered
spaces and strong base conditions," with
H. Bennett, Topology and its Applications
119(2002), 305-314.
pdf version

"Recent developments in the topology of
ordered spaces," with H. Bennett, Recent Progress
in General Topology II, ed. by M. Husek and
J. van Mill, Elsevier, Amsterdam, 2002, pp. 83-110.
pdf version

"Advising," in Encyclopedia of Mathematics
Education, ed. by L. Grinstein and S. Lipsey,
pp. 14-16, RutledgeFalmer, New York, 2001.

"Cleavability in ordered spaces," with
H. Bennett and R. Byerly, ORDER, 18(2001), 1-17.
pdf version

"A decade of REU at William and Mary," with
C. Johnson, Proceedings of the Conference on
Summer Undergraduate Mathematics Research
Programs, ed. by J. Gallian, American Mathematical
Society, Providence, 2000, pp. 19-29.
pdf
version

"OIF spaces," with Z. Balogh, H. Bennett,
D. Burke, G. Gruenhage, and J. Mashburn, Questions
and Answers in General Topology, 18(2000),
129-141. pdf
version

"Generalized ordered spaces with
$\sigma$-closed discrete dense subsets," with
H. Bennett and R. Heath, Proc. Amer. Math. Soc.,
129 (2000), 931-939.
pdf version

"Baire and Volterra spaces," with Gary
Gruenhage, Proc. Amer. Math. Soc., 128 (2000), 3115-3124.
pdf version

"Weakly perfect generalized ordered spaces,"
with H. Bennett and M. Hosobuchi, Houston
J. Math., 26(2000), 609-627.
pdf version

"A note on perfect ordered spaces," with
H. Bennett and M. Hosobuchi, Rocky
Mountain Journal of Mathematics, 29(1999),
1195-1207.
pdf version

"Isomorphic Subgroups and an application of
Q-linearity," with S. Fallat, C-K Li, and D.
Stanford, Mathematics Magazine, 72 (1999),
388-391.
pdf version

"On dense subspaces of generalized ordered
spaces," with H. Bennett and S. Purisch,
Topology and its Applications 93 (1999), 191-205.
pdf version

"Ordered spaces with special bases," with
H. Bennett, Fundamenta Math. 158 (1998), 289-299.
pdf version

"A metric space of A.H. Stone and an example
concerning $\sigma$-minimal bases," with H. Bennett,
Proc. Amer. Math. Soc. 126 (1998), 2183-2189.
pdf version

"Off-diagonal metrization theorems," with
H. Bennett, Topology Proceedings 22 (1997), 37-58.
pdf version

"A note on paracompactness in generalized
ordered spaces," with E. van Douwen,
Proc. Amer. Math. Soc. 125 (1997), 1237-1245.
pdf version

"Diagonal conditions in ordered spaces," with
H. R. Bennett, Fund. Math. 153 (1997), 99-123.
pdf version

"Point-countability in generalized ordered
spaces," with H.R. Bennett, Topology and its
Applications, 71 (1996), 149-165.
pdf version

"A note on Property III in generalized
ordered spaces," with H.R. Bennett, Topology
Proceedings 21 (1996), 15-24.
pdf version

"Advising undergraduates about graduate
school in mathematics: methods that work," with
D. Herrmann, FOCUS, June 1995.

## Papers before 1994 with MR numbers

[1] 88k:54050 Burke, D. K. ; Lutzer, D. J. On powers of certain lines. Topology
Appl. 26 (1987), no. 3, 251--261).
[2] 87e:54046 Lutzer, D. ; van Mill, J. ; Pol, R. Descriptive complexity of
function spaces. Trans. Amer. Math. Soc. 291 (1985), no. 1, 121--128.
[3] 87a:54039 Burke, D. ; Lutzer, D. ; Levi, S. Functional characterizations of
certain $p$-spaces. Topology Appl. 20 (1985), no. 2, 161--165.
[4] 87a:54018 Dijkstra, J. ; Grilliot, T. ; Lutzer, D. ; van Mill, J. Function
spaces of low Borel complexity. Proc. Amer. Math. Soc. 94 (1985), no. 4,
703--710.
[5] 85h:54058 Lutzer, D. J. Twenty questions on ordered spaces. Topology and
order structures, Part 2 (Amsterdam, 1981), 1--18, Math. Centre Tracts, 169,
Math. Centrum, Amsterdam, 1983.
[6] 85h:54053 Bennet, H. ; Lutzer, D. J. A note on perfect normality in
generalized ordered spaces. Topology and order structures, Part 2 (Amsterdam,
1981), 19--22, Math. Centre Tracts, 169, Math. Centrum, Amsterdam, 1983.
[7] 85g:54001 Topology and order structures. Part 2. Papers presented at the
second workshop on topology and linear orderings held in Amsterdam, August
1981. Edited by H. R. Bennett and D. J. Lutzer. Mathematical Centre Tracts,
169. Mathematisch Centrum, Amsterdam, 1983. iii+107 pp. ISBN: 90-6196-266-8
54-06.
[8] 85d:54041 Bennett, H. R. ; Lutzer, D. J. Generalized ordered spaces with
capacities. Pacific J. Math. 112 (1984), no. 1, 11--19.
[9] 83c:54044 Kulpa, W. ; Lutzer, D. New proofs of a metrization theorem for
ordered spaces. Topology and order structures, Part 1 (Lubbock, Tex., 1980),
pp. 141--146, Math. Centre Tracts, 142, Math. Centrum, Amsterdam, 1981.
[10] 83b:54040 Bennett, H. R. ; Lutzer, D. J. A note on perfect ordered spaces.
The Proceedings of the 1980 Topology Conference (Univ. Alabama, Birmingham,
Ala., 1980). Topology Proc. 5 (1980), 27--32 (1981).
[11] 82m:54029 Lutzer, David J. A metrization theorem for generalized ordered
spaces. The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens,
Ohio, 1979). Topology Proc. 4 (1979), no. 1, 265--269 (1980).
[12] 82k:54019 Lutzer, D. J. ; McCoy, R. A. Category in function spaces. I.
Pacific J. Math. 90 (1980), no. 1, 145--168.
[13] 82k:54010 Bennett, H. R. ; Fleissner, W. G. ; Lutzer, D. J.
Ultraparacompactness in certain Pixley\mhy Roy hyperspaces. Fund. Math. 111
(1981), no. 1, 11--18.
[14] 82k:54009 Bennett, H. R. ; Fleissner, W. G. ; Lutzer, D. J. Metrizability
of certain Pixley\mhy Roy spaces. Fund. Math. 110 (1980), no. 1, 51--61.
[15] 82j:54063 Bennett, H. R. ; Lutzer, D. J. Certain hereditary properties and
metrizability in generalized ordered spaces. Fund. Math. 107 (1980), no. 1,
71--84.
[16] 82h:54001 Topology and order structures. Part 1. Papers presented at a
Workshop on Topology and Linear Orderings held at Texas Tech University,
Lubbock, Tex., August 1980. Edited by H. R. Bennett and D. J. Lutzer.
Mathematical Centre Tracts, 142. Mathematisch Centrum, Amsterdam, 1981. iii+184
pp. ISBN: 90-6196-228-5 54-06
[17] 82f:54046 Bennett, H. R. ; Lutzer, D. J. ; van Wouwe, J. M. Linearly
ordered Eberlein compact spaces. Topology Appl. 12 (1981), no. 1, 11--18.
[18] 81d:54026 Lutzer, David J. Ordered topological spaces. Surveys in general
topology, pp. 247--295, Academic Press, New York-London-Toronto, Ont., 1980.
[19] 81b:54008 van Douwen, E. K. ; Lutzer, D. J. ; Pelant, J. ; Reed, G. M. On
unions of metrizable subspaces. Canad. J. Math. 32 (1980), no. 1, 76--85.
[20] 80m:54012 Lutzer, David J. Pixley-Roy topology. Proceedings of the 1978
Topology Conference (Univ. Oklahoma, Norman, Okla., 1978), I. Topology Proc. 3
(1978), no. 1, 139--158 (1979).
[21] 80j:54027 Bennett, H. R. ; Lutzer, D. J. Ordered spaces with $\sigma
$-minimal bases. Proceedings of the 1977 Topology Conference (Louisiana State
Univ., Baton Rouge, La., 1977), II. Topology Proc. 2 (1977), no. 2, 371--382
(1978).
[22] 80h:54038 van Douwen, Eric K. ; Lutzer, David J. On the classification of
stationary sets. Michigan Math. J. 26 (1979), no. 1, 47--64.
[23] 80d:54014 Lutzer, David J. ; Przymusi\'nski, Teodor C. Continuous
extenders in normal and collectionwise normal spaces. Fund. Math. 102 (1979),
no. 3, 165--171.
[24] 56 16577 van Douwen, E. K. ; Lutzer, D. J. ; Przymusi\'nski, T. C. Some
extensions of the Tietze-Urysohn theorem. Amer. Math. Monthly 84 (1977), no. 6,
435--441.
[25] 55 11197 Lutzer, David J. A selection-theoretic approach to certain
extension theorems. Set-theoretic topology (Papers, Inst. Medicine and Math.,
Ohio Univ., Athens, Ohio, 1975--1976), pp. 269--275. Academic Press, New York,
1977.
[26] 55 1318 Burke, Dennis K. ; Lutzer, David J. Recent advances in the theory
of generalized metric spaces. Topology (Proc. Ninth Annual Spring Topology
Conf., Memphis State Univ., Memphis, Tenn., 1975), pp. 1--70. Lecture Notes in
Pure and Appl. Math., Vol. 24, Dekker, New York, 1976.
[27] 55 1303 Engelking, R. ; Lutzer, D. Paracompactness in ordered spaces.
Fund. Math. 94 (1977), no. 1, 49--58.
[28] 54 13844 Franklin, S. P. ; Lutzer, D. J. ; Thomas, B. V. S. On
subcategories of TOP. Trans. Amer. Math. Soc. 225 (1977), 267--278.
[29] 54 6088 Aarts, J. M. ; Lutzer, D. J. On constructing bases in certain
complete spaces. Nederl. Akad. Wetensch. Proc. Ser. A {\bf 79}=Indag. Math. 38
(1976), no. 3, 185--188.
[30] 53 3237 Lutzer, D. J. Some constructions in abstract measure theory.
Canad. J. Math. 27 (1975), no. 4, 860--866.
[31] 52 1642 Aarts, J. M. ; Lutzer, D. J. Completeness properties designed for
recognizing Baire spaces. Dissertationes Math. (Rozprawy Mat.) 116 (1974), 48
pp.
[32] 51 11424 Heath, R. W. ; Lutzer, D. J. The Dugundji extension theorem and
collectionwise normality. Bull. Acad. Polon. Sci Sér. Sci. Math. Astronom.
Phys. 22 (1974), 827--830.
[33] 51 9030 Heath, R. W. ; Lutzer, D. J. ; Zenor, P. L. Monotonically normal
spaces. Trans. Amer. Math. Soc. 178 (1973), 481--493.
[34] 51 6746 Burke, D. ; Engelking, R. ; Lutzer, D. Hereditarily
closure-preserving collections and metrization. Proc. Amer. Math. Soc. 51
(1975), 483--488.
[35] 51 6704 Heath, R. W. ; Lutzer, D. J. Dugundji extension theorems for
linearly ordered spaces. Pacific J. Math. 55 (1974), 419--425.
[36] 50 14692 Lutzer, D. J. Ordinals and paracompactness in ordered spaces.
TOPO 72---general topology and its applications (Proc. Second Pittsburgh
Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H.
de Groot), pp.258--266, Lecture Notes in Math., Vol. 378, Springer, Berlin,
1974.
[37] 50 14681 Borges, C. R. ; Lutzer, D. J. Characterizations and mappings of
$M\sb{i}$ spaces. Topology Conference (Virginia Polytech. Inst. and State
Univ., Blacksburg, Va., 1973), pp. 34--40. Lecture Notes in Math., Vol. 375,
Springer, Berlin, 1974.
[38] 50 11134 Heath, R. W. ; Lutzer, D. J. ; Zenor, P. L. On continuous
extenders. Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte,
N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), pp. 203--213. Academic
Press, New York, 1975.
[39] 50 8460 Lutzer, D. J. On quasi-uniform bases. Proceedings of the
University of Oklahoma Topology Conference dedicated to Robert Lee Moore
(Norman, Okla., 1972), pp. 104--117. Univ. of Oklahoma, Norman, Okla., 1972.
[40] 49 9797 Lutzer, D. ; Martin, H. A note on the Dugundji extension theorem.
Proc. Amer. Math. Soc. 45 (1974), 137--139.
[41] 48 5009 Aarts, J. M. ; Lutzer, D. J. Pseudo-completeness and the product
of Baire spaces. Pacific J. Math. 48 (1973), 1--10.
[42] 48 3018 Lutzer, D. J. On generalized ordered spaces. Dissertationes Math.
Rozprawy Mat. 89 (1971), 32 pp.
[43] 46 8167 Aarts, J. M. ; Lutzer, D. J. The product of totally nonmeagre
spaces. Proc. Amer. Math. Soc. 38 (1973), 198--200.
[44] 46 6292 Lutzer, David J. Another property of the Sorgenfrey line.
Compositio Math. 24 (1972), 359--363.
[45] 46 853 Bennett, H. R. ; Lutzer, D. J. A note on weak $\theta
$-refinability. General Topology and Appl. 2 (1972), 49--54.
[46] 45 5952 Lutzer, D. J. Semimetrizable and stratifiable spaces. General
Topology and Appl. 1 (1971), no. 1, 43--48.
[47] 42#1075 Lutzer, David J. On generalized ordered spaces. 1970 Proc.
Washington State Univ. Conf. on General Topology (Pullman, Wash., 1970) pp.
102--110 Pi Mu Epsilon, Dept. of Math., Washington State Univ., Pullman, Wash.
[48] 40#2013 Lutzer, D. J. ; Bennett, H. R. Separability, the countable chain
condition and the Lindelof property in linearly orderable spaces. Proc. Amer.
Math. Soc. 23 1969 664--667.
[49] 40#2012 Lutzer, David J. A metrization theorem for linearly orderable
spaces. Proc. Amer. Math. Soc. 22 1969 557--558.
[50] 078 649 Bennett, H. R. ; Lutzer, D. J. Problems in perfect ordered spaces.
Open problems in topology, 231--236, North-Holland, Amsterdam, 1990.
[51] 776 568 Bennett, H. R. ; Lutzer, D. J. Metrization, quasi-developments and
$\sigma$-minimal bases. Questions Answers Gen. Topology 2 (1984), no. 2,
73--76.