We consider the space $D(X,Y)$ of densely continuous forms introduced by Hammer and McCoy and investigated also by Holá. We show some additional properties of $D(X,Y)$ and investigate the subspace $D^*(X)$ of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence $\tau _p$. The largest part of the paper is devoted to the study of various cardinal functions for $(D^*(X),\tau _p)$, in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, $\pi $-weight, $\pi $-character, netweight etc.
a1_This study deals with the ethnographic boundary in Moravia, which can be observed in various expressions of folk culture, but which appears most strikingly in folk music. The author does not relinquish this branch of ethnography, although he makes use of the plentiful literature existing both in the other branches of ethnography and also in the field of historical, dialectological and literary research.
In Moravia, roughly in the Eastern third, there runs a fairly shar boundary line which devides the Czechoslovak territory in the field of folk musical culture into two fundamentally distinct stylistic tracts. The boundary can be traced approximately from the south of the Pavlov Heights and the Moravian-Silesian Beskyds. It runs, then, along the Western border of the distinctive East-Moravian ethnographical regions - of Slovácko, Valašsko, Lašsko and Southern Těšín. With regard to this dividing line running through Moravia we speak in this country of the Western type of song, including the song culture of Western Moravia and Bohemia, and the Eastern type of song, covering the melodically much more numerous strata of East-Moravian and Slovakian airs.
There are considerable differences as regards both melody and rhythm between these two stylistic tracts. In Bohemian and West-Moravian songs among the fundamental melodic features are the harmonic scales, especially the clearly prevailing major pattern. The airs are frequently based on an extended triad, they delight in wide intervals and are permanently bound by the basic harmonic functions, while the most characterstic feature of the airs in the oldest stratum of the Eastern type is the pre-harmonic melodic principle and in the numerically strong next layer it is the clash of this principle with the vertical-harmonic outlook. The airs avoid the ther mechanical repetition of melodic ideas, the relationship of the two kinds of scale is roughly equal, or rather an inclination towards the minor can be found almost more frequently. Of the other melodic features of the songs in the Western style we may on the other hand mention the clear periodic and motif structure., a2_A very typical feature too is what is known as the "breaking through" or "embroidery" of the airs; this is the frequent occurrence of a pair of legato notes for which a single syllable exists in the text. Among the basic metrorhythmixcal features in the songs of the Eastern type we find a tendency to irregular construction, the absolute prevalence of the two-four time in dance songs, of course with certain remarkable peculiarities of rhythmization, and of metrical looseness in parlando songs. The song of the Western style is firmer as regards form, more transparent in rhythm and often makes use of three-four or three-eight time. Both the basic metrorhythmical features thus closely complement the features of the melody structure.
The differentiation of the two melodic styles into Western and Eastern finds good support in the geography of the Czechoslovakian territory. However these two terms are not common in this country and are not completely accepted by all investigators. Some of these have already selected the term Instrumental Song Type for the song of Bohemia and Western Moravia and for Eastern Moravia and Slovakia the term Vocal Song Type. These two different concepts are discussed. It is, however, natural that the two stylistic tracts in certain places coincide with each other and that within the two regions further areas have been formed. For example in the territory of Eastern Moravia we also speak of two larger zones or spheres: of the zone directly bound up with the Slovakian, and especially with the West-Slovakian tradition (Slovácko and Valašsko) and the zone more closely bound with the Polish-Silesian tradition (Lašsko and especially the Southern Těšín district). Both these zones are linked by inter-ethnic relationships. Certain general features of the song of the two regions spoken of continue of much further beynd the boundaries of Czechoslovakia. Relationships with the surrounding nations appear, especially with those directly neighbouring to Czechoslovakia. We cannot speak only of connections with Western European and Eastern European culture, as is sometimes done. The relationships are not everywhere simple; in places they are faint, elsewhere striking, clear at the first glance. It is not always a case of East-West, but also of the connection North-South. While the song of the Western type shows a relationship to the culture of the Northern and Western European nations, the old melodic foundation in the Eastern song regions in this country indicates rather connections with Southern Europe and then with the East., a3_The existence of two melodic stylistic tracts on the territory of Czechoslovakia, which clach with each other in Moravia and form one of the basic features of the ethnographic boundary in general, as too the development of certain ethnographic regions, cannot be placed in a one-sided relationship either as regards the oldest periods, or as regards comparatively recent historical facts. That the ethnographical borderline in Moravia did not appear suddenly is best proved by the East-Moravian melodic material itself. It shows the appropriate stylistic relationship with the East and South-East both in the oldest and in the most recent strata. A remarkable fact is that New Hungarian song which was spread throughout Moravia almost without exception only from the end of last century, does not pass beyond the border of the Eastern song type in Moravia. This can be accounted for by general conditions; resulting perhaps not only from the related development of orchestral groupings, but apparently also from the older melodic base. A fertile soil was to be found here for the acceptance also of a new musical culture which from our point of view belongs to the East.
Also it cannot be denied that facts and hypotheses presented permit us to consider an older basis for the ethnographic boundary in Moravia, it is incontrovertible that a number of different elements did not appear previous to the course of later development, with the result that the borderline became still more distinct. In further research it will be necessary to ascertain further links in the chain of development or to cut out some links which are given in certain of the subjects, in order either to make it more firm or else more adequately connected or reconstructed. At the same time it will be necessary not to take as a new starting-point merely the ethnomusicological view-point, or merely the ethnographical one, but it will be essential to carry out a still more profound synthesis of the knowledge of all the specialist branches which have anything to say on the subject., and Článek zahrnuje širší poznámkový aparát