Celebratio Mathematica — Moore

[1]E. Schuyler, J. A. V. Gross, R. L. Moore, and G. B. M. Zerr: “Solutions of problems: Algebra: 123,” Amer. Math. Monthly 8 : 10 (October 1901), pp. 196. MR 1515377 article

[2]R. L. Moore and I. M. DeLong: “Problems for solution: Miscellaneous: 143–144,” Amer. Math. Monthly 11 : 2 (1904), pp. 45. MR 1516082 article

[3]R. L. Moore: Sets of metrical hypotheses for geometry. Ph.D. thesis, University of Chicago, 1905. Advised by O. Veblen and E. H. Moore. See also article with the same title in Trans. Am. Math. Soc. 9:4 (1908). MR 2936795 phdthesis

Oswald Veblen	Related
E. H. Moore	Related	Volume

[4]R. L. Moore: “Geometry in which the sum of the angles of every triangle is two right angles,” Trans. Am. Math. Soc. 8 : 3 (July 1907), pp. 369–378. MR 1500791 JFM 38.0503.02 article

[5]R. L. Moore: “Sets of metrical hypotheses for geometry,” Trans. Am. Math. Soc. 9 : 4 (October 1908), pp. 487–512. See also Moore’s PhD thesis (1905). MR 1500823 JFM 39.0538.06 article

[6]R. L. Moore: “On Duhamel’s theorem,” Ann. Math. (2) 13 : 1–4 (1911–1912), pp. 161–166. MR 1502430 JFM 43.0354.02 article

[7]R. L. Moore: “A note concerning Veblen’s axioms for geometry,” Trans. Am. Math. Soc. 13 : 1 (January 1912), pp. 74–76. MR 1500906 JFM 43.0560.03 article

[8]R. L. Moore: “The linear continuum in terms of point and limit,” Ann. Math. (2) 16 : 1–4 (1914–1915), pp. 123–133. MR 1502498 JFM 45.1219.17 article

[9]R. L. Moore: “On the linear continuum,” Bull. Am. Math. Soc. 22 : 3 (1915), pp. 117–122. MR 1559727 JFM 45.1345.02 article

[10]R. L. Moore: “On a set of postulates which suffice to define a number-plane,” Trans. Am. Math. Soc. 16 : 1 (January 1915), pp. 27–32. MR 1500996 JFM 45.0728.05 article

[11]R. L. Moore: “On the foundations of plane analysis situs,” Trans. Am. Math. Soc. 17 : 2 (1916), pp. 131–164. A brief article with the same title published in Proc. Nat. Acad. Sci. U.S.A. 2:5 (1916). MR 1501033 JFM 46.0828.02 article

[12]R. L. Moore: “Concerning a non-metrical pseudo-Archimedean axiom,” Bull. Am. Math. Soc. 22 : 5 (1916), pp. 225–236. MR 1559766 JFM 45.0728.01 article

[13]R. L. Moore: “On the foundations of plane analysis situs,” Proc. Nat. Acad. Sci. U.S.A. 2 : 5 (May 1916), pp. 270–272. A longer article with the same title was published in Trans. Am. Math. Soc. 17:2 (1916). JFM 46.1498.03 article

[14]R. L. Moore: “A theorem concerning continuous curves,” Bull. Am. Math. Soc. 23 : 5 (1917), pp. 233–236. A related research report was published in Bull. Am. Math. Soc. 23:4 (1917). MR 1559925 JFM 46.0829.05, 46.0308.03 article

[15]R. L. Moore: “A theorem concerning continuous curves,” Bull. Am. Math. Soc. 23 : 4 (1917), pp. 162. Research report only. Article in Bull. Am. Math. Soc. 23:5 (1917). JFM 46.0308.02 article

[16]R. L. Moore: “A characterization of Jordan regions by properties having no reference to their boundaries,” Proc. Nat. Acad. Sci. U.S.A. 4 : 12 (December 1918), pp. 364–370. JFM 46.1451.02 article

Schoenflies [1902] has formulated a set of conditions under which the common boundary of two domains will be a simple closed curve. A different set has been given by J. R. Kline [1918]. Carathéodory [1912-13] has obtained conditions under which the boundary of a single domain will be such a curve. In each of these treatments, however, conditions are imposed

on the boundary itself,
regarding the relation of the boundary to the domain or domains in question.

In the present paper I propose to establish the following theorem in which all the conditions imposed are on the domain \( R \) alone.

In order that a simply connected, limited, two-dimensional domain \( R \) should have a simple closed curve as its boundary it is necessary and sufficient that \( R \) should be uniformly connected im kleinen.

[17]R. L. Moore: “On the Lie–Riemann-Helmholtz–Hilbert problem of the foundations of geometry,” Am. J. Math. 41 : 4 (October 1919), pp. 299–319. MR 1506395 JFM 47.0513.02 article

[18]R. L. Moore: “Concerning a set of postulates for plane analysis situs,” Trans. Am. Math. Soc. 20 : 2 (April 1919), pp. 169–178. MR 1501119 JFM 47.0519.06 article

[19]R. L. Moore: “Continuous sets that have no continuous sets of condensation,” Bull. Am. Math. Soc. 25 : 4 (1919), pp. 174–176. MR 1560170 JFM 47.0177.03 article

[20]R. L. Moore: “On the most general class \( L \) of Fréchet in which the Heine–Borel–Lebesgue theorem holds true,” Proc. Nat. Acad. Sci. U.S.A. 5 : 6 (June 1919), pp. 206–210. JFM 47.0896.01 article

[21]R. L. Moore and J. R. Kline: “On the most general plane closed point-set through which it is possible to pass a simple continuous arc,” Ann. Math. (2) 20 : 3 (March 1919), pp. 218–223. MR 1502556 JFM 46.0829.02 article

[22]R. L. Moore: “Book review: ‘Projective geometry’, vol. 2,” Bull. Am. Math. Soc. 26 : 9 (1920), pp. 412–425. Book by O. Veblen and J. W. Young (Ginn & Co., 1918). MR 1560330 JFM 47.0583.01 article

John Wesley Young	Related
Oswald Veblen	Related

[23]R. L. Moore: “Concerning simple continuous curves,” Trans. Am. Math. Soc. 21 : 3 (July 1920), pp. 333–347. MR 1501148 JFM 47.0519.08 article

[24]R. L. Moore: “Concerning certain equicontinuous systems of curves,” Trans. Am. Math. Soc. 22 : 1 (January 1921), pp. 41–55. MR 1501158 JFM 48.0661.01 article

[25]R. L. Moore: “Conditions under which one of two given closed linear point sets may be thrown into the other by continous transformation of a plane into itself,” Bull. Am. Math. Soc. 27 : 7 (1921), pp. 297–298. Abstract only. Article in Am. J. Math. 48:1 (1926). JFM 48.0666.05 article

[26]R. L. Moore: “Concerning continuous curves in the plane,” Math. Z. 15 : 1 (1922), pp. 254–260. MR 1544571 JFM 48.0660.01 article

[27]R. L. Moore: “Concerning connectedness im kleinen and a related property,” Fund. Math. 3 (1922), pp. 232–237. JFM 48.0659.03 article

Sierpiński has recently shown [1920] that in order that a closed and connected set of points \( M \) should be a continuous curve it is necessary and sufficient that, for every positive number \( \varepsilon \), the point-set \( M \) should be the sum of a finite number of closed and connected point-sets each of diameter less than \( \varepsilon \). It follows that, as applied to point-sets which are closed, bounded and connnected, this property is equivalent to that of connectedness im kleinen. In the present paper I will make a further study of these two properties (or rather of suitable modifications of these properties) especially as applied to sets which are not necessarily closed.

[28]R. L. Moore: “On the relation of a continuous curve to its complementary domains in space of three dimensions,” Proc. Nat. Acad. Sci. U.S.A. 8 : 3 (March 1922), pp. 33–38. JFM 48.0660.02 article

[29]R. L. Moore: “Concerning relatively uniform convergence,” Bull. Am. Math. Soc. 28 : 6 (1922), pp. 293. Abstract only. Article in Bull. Am. Math. Soc. 30:9–10 (1924). JFM 48.0299.05 article

According to E. H. Moore, a sequence of functions \( f_1(p) \), \( f_2(p) \), \( f_3(p),\dots \), defined on a range \( K \), is said to converge, to a function \( f(p) \), relatively uniformly with respect to the scale function \( s(p) \) if, for every positive number \( e \), there exists a positive number \( \delta_e \) such that if \( n > \delta_e \) then, for every \( p \) which belongs to \( K \), \[ |f_n(p)-f(p)| < e|s(p)| .\] In this note I will establish the following theorem: If \( S \) is a convergent sequence of measurable functions \( f_1(x) \), \( f_2(x) \), \( f_3(x),\dots \) defined on a measurable point set \( E \) and \( S \) converges for each \( x \) belonging to \( E \), then \( E \) contains a subset \( E_0 \) of measure zero such that the sequence \( S \) converges relatively uniformly for all values of \( x \) on the range \( E-E_0 \).

[30]R. L. Moore: “Report on continuous curves from the viewpoint of analysis situs,” Bull. Am. Math. Soc. 29 : 7 (1923), pp. 289–302. MR 1560734 JFM 49.0401.03 article

[31]R. L. Moore: “Concerning the cut-points of continuous curves and of other closed and connected point-sets,” Proc. Nat. Acad. Sci. U.S.A. 9 : 4 (April 1923), pp. 101–106. JFM 49.0143.01 article

[32]R. L. Moore: “On the generation of a simple surface by means of a set of equicontinuous curves,” Fund. Math. 4 (1923), pp. 106–117. JFM 49.0402.01 article

[33]R. L. Moore: “An uncountable, closed, and non-dense point set each of whose complementary intervals abuts on another one at each of its ends,” Bull. Am. Math. Soc. 29 : 2 (1923), pp. 49–50. MR 1560661 JFM 49.0142.04 article

[34]R. L. Moore: “A connected and regular point set which contains no arc,” Bull. Am. Math. Soc. 29 : 10 (1923), pp. 438. Abstract only. Article in Bull. Am. Math. Soc. 32:4 (1926). JFM 49.0412.16 article

[35]R. L. Moore: “Concerning the sum of a countable number of mutually exclusive continua in the plane,” Fund. Math. 6 (1924), pp. 189–202. JFM 50.0137.04 article

[36]R. L. Moore: “An extension of the theorem that no countable point set is perfect,” Proc. Nat. Acad. Sci. U.S.A. 10 : 5 (May 1924), pp. 168–170. JFM 50.0130.01 article

[37]R. L. Moore: “Concerning upper semi-continuous collections of continua which do not separate a given continuum,” Proc. Nat. Acad. Sci. U.S.A. 10 : 8 (August 1924), pp. 356–360. An abstract was published in Bull. Am. Math. Soc. 31:3 (1925). JFM 50.0130.04 article

In her thesis [1922], Miss Anna M. Mullikin showed that if \( M \) is the set ofall points in a given plane and \( G \) is a countable collection of mutually exclusive closed subsets of \( M \) no one of which disconnects \( M \) then \( M \) is not disconnected by the sum of all the point sets of the collection \( G \). It is clear that this proposition does not remain true if the phrase “\( M \) is the set of all points in a given plane” is replaced by the phrase “\( M \) is a closed and connected subset of the set \( S \) of all the points of a given plane.” If \( M \) is any simple closed curve, \( A \) and \( B \) are two points belonging to \( M \) and the point sets of the collection \( G \) are the points \( A \) and \( B \) then no point set of the collection \( G \) separates \( M \), but their sum does separate \( M \). It is to be observed that in this case \( M \) disconnects \( S \).

[38]R. L. Moore: “Concerning the common boundary of two domains,” Fund. Math. 6 (1924), pp. 203–213. JFM 50.0138.04 article

In an article [1924] published in Volume 5 of this journal, J. R. Kline has shown that in order that a bounded continuum should be a simple closed curve it is necessary and sufficient that it should remain connected (in the weak sense) on the removal of any one of its connected proper subsets. In the same volume C. Kuratowski [1924] has shown that in order that a bounded continuum should be a simple closed curve it is necessary and sufficient that it should remain connected in the strong sense on the removal of any one of its connected proper sets which is closed. In the present paper I will show, among other things, that if a bounded continuum has more than one prime part and no one of its prime parts separates the plane then in order that it should have just two complementary domains and be the complete boundary of each of them it is necessary and sufficient that it should remain connected in the weak sense on the removal of any one of its connected proper subsets which is closed.

[39]R. L. Moore: “Concerning relatively uniform convergence,” Bull. Am. Math. Soc. 30 : 9–10 (1924), pp. 504–505. An abstract was published in Bull. Am. Math. Soc. 28:6 (1922). MR 1560954 JFM 50.0179.01 article

According to E. H. Moore, a sequence of functions \( f_1(p) \), \( f_2(p) \), \( f_3(p),\dots \), defined on a range \( K \), is said to converge, to a function \( f(p) \), relatively uniformly with respect to the scale function \( s(p) \) if, for every positive number \( e \), there exists a positive number \( \delta_e \) such that if \( n > \delta_e \) then, for every \( p \) which belongs to \( K \), \[ |f_n(p)-f(p)| < e|s(p)| .\] In this note I will establish the following theorem: If \( S \) is a convergent sequence of measurable functions \( f_1(x) \), \( f_2(x) \), \( f_3(x),\dots \) defined on a measurable point set \( E \) and \( S \) converges for each \( x \) belonging to \( E \), then \( E \) contains a subset \( E_0 \) of measure zero such that the sequence \( S \) converges relatively uniformly for all values of \( x \) on the range \( E-E_0 \).

[40]R. L. Moore: “Concerning sets of segments which cover a point set in the Vitali sense,” Proc. Nat. Acad. Sci. U.S.A. 10 : 11 (November 1924), pp. 464–467. An abstract was published in Bull. Am. Math. Soc. 31:3 (1925). JFM 50.0131.01 article

A set of segments \( G \) will be said to cover a point set \( K \) in the Vitali sense if, for every point \( P \) which belongs to \( K \) and every positive number \( e \), there exists a segment of \( G \) which contains \( P \) and is of length less than \( e \). In a recent paper [1924] J. Spława-Neyman has shown that if, in space of one dimension, \( K \) is a closed and bounded point set of measure zero and \( G \) is a set of segments which covers \( K \) in the Vitali sense then for every positive number \( e \) the set \( G \) contains a subject \( G_e \), such that \( G_e \), covers \( K \) and such that the sum of the lengths of all the segments of the set \( G_e \) is less than \( e \). He indicates that Sierpinski has raised the question whether this theorem remains true if the stipulation that \( K \) be closed is omitted. I will show that this question may be answered in the negative.

[41]R. L. Moore: “Concerning the prime parts of certain continua which separate the plane,” Proc. Nat. Acad. Sci. U.S.A. 10 : 5 (May 1924), pp. 170–175. JFM 50.0130.02 article

[42]R. L. Moore: “A characterization of a continuous curve,” Bull. Am. Math. Soc. 30 : 9 (1924), pp. 484. Abstract only. Article in Fund. Math. 7 (1925). JFM 50.0374.07 article

[43]R. L. Moore: “Concerning upper semi-continuous collections of continua,” Trans. Am. Math. Soc. 27 : 4 (October 1925), pp. 416–428. MR 1501320 JFM 51.0464.03 article

[44]R. L. Moore: “Concerning the separation of point sets by curves,” Proc. Nat. Acad. Sci. U.S.A. 11 : 8 (August 1925), pp. 469–476. An abstract was published in Bull. Am. Math. Soc. 32:1 (1926). JFM 51.0459.04 article

[45]R. L. Moore: “Concerning the prime parts of a continuum,” Math. Z. 22 : 1 (1925), pp. 307–315. MR 1544726 JFM 51.0461.02 article

[46]R. L. Moore: “A characterization of a continuous curve,” Fund. Math. 7 (1925), pp. 302–307. An abstract was published in Bull. Am. Math. Soc. 30:9 (1924). JFM 51.0462.04 article

If \( A \) and \( B \) are points belonging to a connected point set \( M \), a point set \( K \) is said to separate \( A \) from \( B \) in \( M \) if \( K \) is a proper subset of \( M \) and \( M - K \) is the sum of two mutually separated point sets which contain \( A \) and \( B \) respectively. It is shown that in order that a plane continuum \( M \) shall be a continuous curve it is necessary and sufficient that, for every two points \( A \) and \( B \) that belong to \( M \), there exist a finite number of continua whose sum separates \( A \) from \( B \) in \( M \). In order that a bounded plane continuum \( M \) shall be a continuous curve which neither contains a domain nor separates the plane, it is necessary and sufficient that, for every two distinct points \( A \) and \( B \) which belong to \( M \), there exist a point which separates \( A \) from \( B \) in \( M \).

[47]R. L. Moore: “Concerning sets of segments which cover a point set in the Vitali sense,” Bull. Am. Math. Soc. 31 : 3 (1925), pp. 120. Abstract only. Article in Proc. Nat. Acad. Sci. U.S.A. 10:11 (1924). JFM 51.0170.11 article

[48]R. L. Moore: “Covering theorems,” Bull. Am. Math. Soc. 31 : 5 (1925), pp. 219–220. Abstract only. Article in Bull. Am. Math. Soc. 32:3 (1926). JFM 51.0170.09 article

[49]R. L. Moore: “Concerning upper semi-continuous collections,” Bull. Am. Math. Soc. 31 : 5 (1925), pp. 220. Early research report published in Monatsh. Math. Phys. 36:1 (1929). See also Bull. Am. Math. Soc. 34:4 (1928). JFM 51.0466.10 article

[50]R. L. Moore: “Concerning upper semi-continuous collections of continua which do not separate a given continuum,” Bull. Am. Math. Soc. 31 : 3 (1925), pp. 120. Abstract only. Article in Proc. Nat. Acad. Sci. U.S.A. 10:8 (1924). JFM 51.0466.08 article

[51]R. L. Moore: “Concerning the relation between separability and the proposition that every uncountable point set has a limit point,” Fund. Math. 8 (1926), pp. 189–192. An abstract was published in Bull. Am. Math. Soc. 32:1 (1926). JFM 52.0583.03 article

[52]R. L. Moore: “Concerning indecomposable continua and continua which contain no subsets that separate the plane,” Proc. Nat. Acad. Sci. U.S.A. 12 : 5 (May 1926), pp. 359–363. JFM 52.0600.04 article

[53]R. L. Moore: “Conditions under which one of two given closed linear point sets may be thrown into the other by continous transformation of a plane into itself,” Am. J. Math 48 : 1 (January 1926), pp. 67–72. An abstract was published in Bull. Am. Math. Soc. 27:7 (1921). MR 1506573 JFM 52.0604.03 article

[54]R. L. Moore: “Covering theorems,” Bull. Am. Math. Soc. 32 : 3 (1926), pp. 275–282. An abstract was published in Bull. Am. Math. Soc. 31:5 (1925). MR 1561204 JFM 52.0197.03 article

Using the following definition: A point set \( K \) is said to be uniformly relatively well imbedded in a set of regions \( G \) if there exists a positive number \( d \), less than unity, such that each point of \( K \) belongs to some region \( R \) of \( G \) and is at a distance from the boundary of \( R \) less than \( d \) times the diameter of \( R \), the author proves these theorems: (1) If, for each circle \( g \) of a set \( G \), \( h_g \) is a definite circle concentric with \( g \) and there exists a positive \( e \), less than unity, such that, for every \( g \), the radius of \( h_g \) is less than \( e \) times that of \( g \), then the set \( G \) contains a subset such that, for each \( g \), \( h_g \) is within one and only one circle of this subset. (2) If \( G \) is a set of circular regions and \( K \) is a point set of measure zero, and, for every positive \( e \), \( K \) is uniformly relatively well imbedded in a subset of \( G \) whose regions are of radius less than \( e \), then, for every positive \( \varepsilon \), \( G \) contains a subset \( G^* \) which covers \( K \), the sum of the areas of the regions of \( G^* \) being less than \( \varepsilon \). (3) J. Splawa-Neyman’s theorem [1924] remains true if the words “à son intérieur” are omitted.

[55]R. L. Moore: “A connected and regular point set which contains no arc,” Bull. Am. Math. Soc. 32 : 4 (1926), pp. 331–332. An abstract was published in Bull. Am. Math. Soc. 29:10 (1923). MR 1561220 JFM 52.0600.03 article

[56]R. L. Moore: “Concerning paths that do not separate a given continuous curve,” Proc. Nat. Acad. Sci. U.S.A. 12 : 12 (December 1926), pp. 745–753. JFM 52.0603.03 article

[57]R. L. Moore: “Concerning the relation between separability and the proposition that every uncountable point set has a limit point,” Bull. Am. Math. Soc. 32 : 1 (1926), pp. 13. Abstract only. Article in Fund. Math. 8 (1926). JFM 52.0605.14 article

[58]R. L. Moore: “Some separation theorems,” Proc. Nat. Acad. Sci. U.S.A. 13 : 10 (October 1927), pp. 711–716. JFM 53.0564.02 article

[59]R. L. Moore: “A separation theorem,” Bull. Am. Math. Soc. 33 : 6 (1927), pp. 656. Abstract only. Article in Fund. Math. 12 (1928). JFM 53.0574.06 article

[60]R. L. Moore: “On the separation of the plane by a continuum,” Bull. Am. Math. Soc. 34 : 3 (1928), pp. 303–306. MR 1561553 JFM 54.0630.02 article

In his paper [1927], W. A. Wilson obtains the following theorem. Let \( F \) be the union of two bounded continua \( H_1 \) and \( H_2 \) having these properties:

\( H_1 \) and \( H_2 \) are irreducible about \( A+B \);
\( H_1\cdot H_2 = A+B \) where \( A \) and \( B \) are continua and \( A\cdot B = 0 \);
\( H_1 \) and \( H_2 \) contain subcontinua \( C_1 \) and \( C_2 \) respectively such that \[ \alpha = C_1\cdot C_2\cdot A \neq 0, \qquad \beta = C_1\cdot C_2\cdot B\neq 0 ,\] \( C_1 \) and \( C_2 \) are irreducible between \( \alpha \) and \( \beta \), and \( F = C_1 + C_2 \).

Then \( F \) cuts the plane and is the frontier of exatly two components of its complement. In the present paper I will establish two theorems which together yield more information than Wilson’s theorem.

[61]R. L. Moore: “A separation theorem,” Fund. Math. 12 (1928), pp. 295–297. An abstract was published in Bull. Am. Math. Soc. 33:6 (1927). JFM 54.0630.01 article

In this paper the following theorems are proved. (I) If, in a plane \( S \), \( M \) is a bounded continuous curve, \( K \) is a closed subset of \( M \), and \( H \) is a subset of \( K \) and of some connected subset of \( K + (S - M) \) but no point set of which \( H \) is a proper subset satisfies these conditions, then if the continuum \( E \) is a subset of \( K - H \), there exists a simple closed curve lying wholly in \( M - K \) and separating \( H \) from \( E \). (II) If, in a plane \( S \), \( K \) is a closed subset of a continuous curve \( M \), and \( A \) and \( B \) are two points of \( M \), then if \( A \) and \( B \) lie in a connected subset of \( K + (S - M) \) they lie in a closed and connected subset of \( K + (S - M) \). (III) In order that a bounded continuum \( M \) in a plane \( S \) should be a continuous curve, it is necessary and sufficient that if \( t \) is an arc and \( H \) is an open subset of \( M + t \), and \( A \) and \( B \) are points of \( S - H \) which are weakly separated by \( H \), then \( H \) contains a simple closed curve that separates \( A \) from \( B \). The first of these theorems may be regarded as an extension of a theorem of Zoretti’s.

[62]R. L. Moore: “Concerning triods in the plane and the junction points of plane continua,” Proc. Nat. Acad. Sci. U.S.A. 14 : 1 (January 1928), pp. 85–88. An abstract was published in Bull. Am. Math. Soc. 34:2 (1928). JFM 54.0630.03 article

[63]R. L. Moore: “Concerning upper semi-continuous collections,” Bull. Am. Math. Soc. 34 : 4 (1928), pp. 416–417. Abstract only. Article published in Monatsh. Math. Phys. 36:1 (1929). See also Bull. Am. Math. Soc. 31:5 (1925). JFM 54.0645.07 article

[64]R. L. Moore: “Concerning triods in the plane and the junction points of plane continua,” Bull. Am. Math. Soc. 34 : 2 (1928), pp. 137. Abstract only. Article in Proc. Nat. Acad. Sci. U.S.A. 14:1 (1928). JFM 54.0645.04 article

[65]R. L. Moore: “Concerning upper semi-continuous collections,” Monatshefte für Mathematik und Physick 36 : 1 (1929), pp. 81–88. An abstract was published in Bull. Am. Math. Soc. 34:4 (1928) and an early research report in Bull. Am. Math. Soc. 31:5 (1925). MR 1549673 JFM 55.0317.04 article

[66]R. L. Moore: “Concerning triodic continua in the plane,” Fund. Math. 13 (1929), pp. 261–263. JFM 55.0978.03 article

[67]R. L. Moore: Foundations of point set theory. AMS Colloquium Publications 13. American Mathematical Society (Providence, RI), 1932. Revised and enlarged edition published in 1962. JFM 58.0637.02 Zbl 0005.05403 book

[68]R. L. Moore: “Concerning compact continua which contain no continuum that separates the plane,” Bull. Am. Math. Soc. 39 : 9 (1933), pp. 671. Abstract only. Article in Proc. Nat. Acad. Sci. U.S.A. 20:1 (1934). JFM 59.0579.16 article

[69]R. L. Moore: “Concerning compact continua which contain no continuum that separates the plane,” Proc. Nat. Acad. Sci. U.S.A. 20 : 1 (January 1934), pp. 41–45. An abstract was published in Bull. Am. Math. Soc. 39:9 (1933). Zbl 0008.32603 article

It has been shown, by Mazurkiewicz [1921], that every arc which lies in a compact acyclic continuous curve \( M \) is a subset of an arc which lies in \( M \) but which is itself not a proper subset of any other such arc. In the present paper it will be shown that if, in a plane \( \alpha \), \( S \) is a compact continuum containing no continuum that separates \( \alpha \) and \( K \) is a subcontinuum of \( S \) which is irreducible between some two points, then \( K \) is a subset of some subcontinuum of \( S \) which is irreducible between some two points but which is itself not a proper subset of any other such subcontinuum of \( S \). With the aid of this result it will be proved that if, in the plane \( \alpha \), \( M \) is a non-degenerate, compact and atriodic continuum which does not separate \( \alpha \), then \( M \) is irreducible between some two points.

[70]R. L. Moore: “A set of axioms for plane analysis situs,” Fund. Math. 25 (1935), pp. 13–28. JFM 61.0638.04 Zbl 0011.27501 article

[71]R. L. Moore: Fundamental theorems concerning point sets. The Rice Institute Pamphlet 23. January 1936. An elaboration and extension of material presented in a series of three lectures delivered at the Rice Institute. JFM 62.1412.03 booklet

[72]R. L. Moore: “Concerning essential continua of condensation,” Bull. Am. Math. Soc. 42 : 1 (1936), pp. 35–36. Abstact only. Article in Trans. Am. Math. Soc. 42:1 (1936). JFM 62.0696.08 article

[73]R. L. Moore: “Concerning essential continua of condensation,” Trans. Am. Math. Soc. 42 : 1 (July 1937), pp. 41–52. An abstract was published in Bull. Am. Math. Soc. 42:1 (1936). MR 1501913 JFM 63.1165.03 Zbl 0017.09201 article

The continuum \( M \) is said to have property \( N \) if for every positive number \( e \) there exists a finite set \( H \) of non-degenerate subcontinua of \( M \) such that every subcontinuum of \( M \) of diameter more than \( e \) contains some continuum of \( H \). The author has shown (see a forthcoming issue of the Rice Institute Pamphlet [1936]) that no compact continuum with property \( N \) has an essential continuum of condensation. It is now shown that if the compact and webless continuum \( M \) does not have property \( N \) there exists an upper semi-continuous collection of mutually exclusive continua filling up \( M \) and forming, with respect to its elements, a continuum with an essential continuum of condensation. In order that the compact and webless continuum \( M \) should have no essential continuum of condensation it is necessary and sufficient that every non-degenerate subcontinuum of \( M \) should contain uncountably many local separating points of \( M \). In order that such a continuum should have an essential continuum of condensation it is necessary and sufficient that there should exist a positive number \( d \) such that if \( G is an upper semi-continuous collection of mutually exclusive continua filling up \)M\( and forming, with respect to its elements, a continuum with no continuum of condensation, then some continuum of \)G\( is of diameter more than \)d\( . \)

[74]R. L. Moore: “Concerning accessibility,” Proc. Nat. Acad. Sci. U.S.A. 25 : 12 (1 1939), pp. 648–653. MR 0000643 JFM 65.0874.01 Zbl 0023.11401 article

[75]R. L. Moore: “Concerning the open subsets of a plane continuum,” Proc. Nat. Acad. Sci. U.S.A. 26 : 1 (January 1940), pp. 24–25. MR 0000642 JFM 66.0970.02 Zbl 0023.11505 article

[76]R. L. Moore: “Concerning domains whose boundaries are compact,” Proc. Nat. Acad. Sci. U.S.A. 28 : 12 (December 1942), pp. 555–561. MR 0007483 Zbl 0060.40305 article

[77]R. L. Moore: “Concerning a continuum and its boundary,” Proc. Nat. Acad. Sci. U.S.A. 28 : 12 (December 1942), pp. 550–555. MR 0007482 Zbl 0060.40304 article

[78]R. L. Moore: “Concerning intersecting continua,” Proc. Nat. Acad. Sci. U.S.A. 28 (December 1942), pp. 544–550. MR 0007481 Zbl 0060.40303 article

[79]R. L. Moore: “Concerning separability,” Proc. Nat. Acad. Sci. U.S.A. 28 : 2 (February 1942), pp. 56–58. MR 0005320 Zbl 0063.04086 article

[80]R. L. Moore: “Concerning continua which have dendratomic subsets,” Proc. Nat. Acad. Sci. U.S.A. 29 : 11 (December 1943), pp. 384–389. MR 0009435 Zbl 0060.40306 article

[81]R. L. Moore: “Concerning webs in the plane,” Proc. Nat. Acad. Sci. U.S.A. 29 : 11 (December 1943), pp. 389–393. MR 0009436 Zbl 0060.40307 article

[82]R. L. Moore: “Concerning tangents to continua in the plane,” Proc. Nat. Acad. Sci. U.S.A. 31 : 2 (February 1945), pp. 67–70. MR 0011542 Zbl 0063.04085 article

[83]R. L. Moore: “A characterization of a simple plane web,” Proc. Nat. Acad. Sci. U.S.A. 32 : 12 (December 1946), pp. 311–316. MR 0018809 Zbl 0060.40308 article

[84]R. L. Moore: “Spirals in the plane,” Proc. Nat. Acad. Sci. U.S.A. 39 : 3 (March 1953), pp. 207–213. MR 0053496 Zbl 0050.38701 article

[85]R. L. Moore: Foundations of point set theory, revised and enlarged edition. AMS Colloquium Publications 13. American Mathematical Society (Providence, RI), 1962. Republication of 1932 original. MR 0150722 Zbl 0192.28901 book

[86]F. B. Jones: “Dedication,” pp. iv–x in Topology conference (Emory University, Atlanta, GA, 19–21 March 1970). Edited by J. W. Rogers, Jr. Emory University (Atlanta, GA), 1970. Conference dedicated to Robert Lee Moore. MR 0342343 incollection

Floyd Burton Jones	Related
Jack W. Rogers, Jr.	Related

[87]Topology conference (Emory University, Atlanta, GA, 19–21 March 1970). Edited by J. W. Rogers, Jr. Emory University (Atlanta, GA), 1970. Conference dedicated to Robert Lee Moore. MR 0331297 Zbl 0245.00003 book

[88]Proceedings of the University of Oklahoma topology conference (Norman, OK, 23–25 March 1972). Edited by D. Kay, J. Green, L. Rubin, and L. P. Su. University of Oklahoma (Norman, OK), 1972. Conference dedicated to Robert Lee Moore. MR 0341363 Zbl 0237.00017 book

John William Green	Related
David C. Kay	Related
Leonard Roy Rubin	Related
Li Pi Su	Related

[89]D. R. Traylor: “Dedication,” pp. i–xx in Proceedings of the University of Oklahoma topology conference (Norman, OK, 23–25 March 1972). Edited by D. Kay, J. Green, L. Rubin, and L. P. Su. University of Oklahoma (Norman, OK), 1972. Conference dedicated to Robert Lee Moore. MR 0342344 incollection

Donald Reginald Traylor	Related
John William Green	Related
David C. Kay	Related
Leonard Roy Rubin	Related
Li Pi Su	Related

[90]R. L. Wilder: “Robert Lee Moore, 1882–1974,” Bull. Am. Math. Soc. 82 : 3 (1976), pp. 417–427. MR 0465747 Zbl 0329.01016 article

[91]F. B. Jones: “The Moore method,” Am. Math. Monthly 84 : 4 (1977), pp. 273–278. MR 0426988 Zbl 0351.00029 article

[92]R. L. Wilder: “The mathematical work of R. L. Moore: Its background, nature and influence,” Arch. Hist. Exact Sci. 26 : 1 (1982), pp. 73–97. MR 664470 Zbl 0491.01012 article

[93]B. Fitzpatrick, Jr.: “Some aspects of the work and influence of R. L. Moore,” pp. 41–61 in Handbook of the history of general topology, vol. 1. Edited by C. E. Aull and R. Lowen. Kluwer Academic (Dordrecht), 1997. MR 1617585 Zbl 0899.54003 incollection

Charles E. Aull	Related
Ben Fitzpatrick, Jr.	Related
Robert Lowen	Related

[94]J. Parker: R. L. Moore: Mathematician and teacher. MAA Spectrum. Mathematical Association of America (Washington, DC), 2005. MR 2103548 Zbl 1069.01007 book

[95]L. Corry: “A clash of mathematical titans in Austin: Harry S. Vandiver and Robert Lee Moore (1924–1974),” Math. Intelligencer 29 : 4 (2007), pp. 62–74. MR 2361623 Zbl 0554.5034 article

Leo Corry	Related
Harry Schultz Vandiver	Related

Complete Bibliography