Uniform connectedness and uniform local connectedness for lattice-valued uniform convergence spaces

Document Type: Research Paper


School of Mechanical Engineering, University of Applied Sciences Stralsund, 18435 Stralsund, Germany


We apply Preuss' concept of $mbbe$-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly $mbbe$-connected if the only uniformly continuous mappings from the space to a space in the class $mbbe$ are the constant mappings. We develop the basic theory for $mbbe$-connected sets, including the product theorem. Furthermore, we define and study uniform local $mbbe$-connectedness, generalizing a classical definition from the theory of uniform convergence spaces to the lattice-valued case. In particular it is shown that if the underlying lattice is completely distributive, the quotient space of a uniformly locally $mbbe$-connected space and products of locally uniformly $mbbe$-connected spaces are locally uniformly $mbbe$-connected.


[1] J. Adamek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories, Wiley, New
York, 1989.

[2] G. Cantor,  Uber unendliche lineare Punktmannichfaltigkeiten, Math. Ann., 21 (1883), 545{
[3] A. Craig and G. Jager, A common framework for lattice-valued uniform spaces and probabilistic
uniform limit spaces, Fuzzy Sets and Systems, 160 (2009), 1177 { 1203.
[4] J. Fang, Lattice-valued semiuniform convergence spaces, Fuzzy Sets and Systems, 195 (2012),
[5] W. Gahler, Grundstrukturen der Analysis I, Birkhauser Verlag, Basel and Stuttgart, 1977.
[6] J. Gutierrez Garca, A uni ed approach to the concept of fuzzy L-uniform space, Thesis,
Universidad del Pais Vasco, Bilbao, Spain, 2000.
[7] J. Gutierrez Garca, M.A. de Prada Vicente and A. P. Sostak, A uni ed approach to the
concept of fuzzy L-uniform space, In: S. E. Rodabaugh, E. P. Klement (Eds.), Topological
and algebraic structures in fuzzy sets, Kluwer, Dordrecht, (2003), 81{114.
[8] F. Hausdor , Grundzuge der Mengenlehre, Leipzig, 1914.
[9] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, In: U. Hohle,
S.E. Rodabauch (Eds.), Mathematics of Fuzzy Sets. Logic, Topology and Measure Theory,
Kluwer, Boston/Dordrecht/London (1999), 123{272.
[10] G. Jager, A category of L-fuzzy convergence spaces, Quaestiones Math., 24 (2001), 501{517.
[11] G. Jager, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156
(2005), 1{24.
[12] G. Jager and M. H. Burton, Strati ed L-uniform convergence spaces, Quaest. Math., 28
(2005), 11 { 36.
[13] G. Jager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159
(2008), 2488{2502.
[14] G. Jager, Level spaces for lattice-valued uniform convergence spaces, Quaest. Math., 31
(2008), 255{277.
[15] G. Jager, Compactness in lattice-valued function spaces, Fuzzy Sets and Systems, 161 (2010),
[16] G. Jager, Lattice-valued Cauchy spaces and completion, Quaest. Math., 33 (2010), 53{74.
[17] G. Jager, Connectedness and local connectedness for lattice-valued convergence spaces, Fuzzy
Sets and Systems, to appear, doi:10.1016/j.fss.2015.11.013.
[18] G. Kneis, Contributions to the theory of pseudo-uniform spaces, Math. Nachrichten, 89
(1979), 149{163.
[19] S. G. Mrowka and W. J. Pervin, On uniform connectedness, Proc. Amer. Math. Soc., 15
(1964), 446{449.
[20] G. Preu, E-Zusammenhangende Raume, Manuscripta Mathematica, 3 (1970), 331{342.
[21] G. Preu, Trennung und Zusammenhang, Monatshefte fur Mathematik, 74(1970), 70{87.
[22] W. W. Taylor, Fixed-point theorems for nonexpansive mappings in linear topological spaces,
J. Math. Anal. Appl., 40 (1972), 164{173.
[23] R. Vainio, A note on products of connected convergence spaces, Acta Acad. Aboensis, Ser.
B, 36(2) (1976), 1{4.
[24] R. Vainio, The locally connected and the uniformly locally connected core
ector in general
convergence theory, Acta Acad. Aboensis, Ser. B, 39(1) (1979), 1{13.
[25] R. Vainio, On connectedness in limit space theory, in: Convergence structures and applications
II, Abhandlungen der Akad. d. Wissenschaften der DDR, Berlin (1984), 227{232.