Some cardinal functions on algebras II

1975 ◽  
Vol 5 (1) ◽  
pp. 361-366 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Cardinal Functions

Some notes on densely continuous forms

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Peter Vadovič
Keyword(s):  
Acta Math ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Graph Topology ◽  
Uniform Topology ◽  
Pointwise Topology ◽  
Topology Of Pointwise Convergence ◽  
Nonempty Compact ◽  
Cardinal Functions ◽  
Special Space

AbstractWe consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLÁ, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLÝ, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLÝ, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D p*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D p*(X) and on real-valued continuous functions C p(X) and a generalization of a sufficient condition for the countable cellularity of D p*(X).

Cardinal Functions for k-Spaces

1978 ◽  
Vol 68 (3) ◽  
pp. 355 ◽  
Author(s):  
James R. Boone ◽  
Sheldon W. Davis ◽  
Gary Gruenhage
Keyword(s):  
Cardinal Functions

Cardinal functions on ultra products of Boolean algebras

1997 ◽  
Vol 62 (1) ◽  
pp. 43-59 ◽  
Author(s):  
Douglas Peterson
Keyword(s):  
Boolean Algebra ◽  
Cardinal Number ◽  
Boolean Algebras ◽  
Open Problems ◽  
Cardinal Functions ◽  
Image Position

This article is concerned with functions k assigning a cardinal number to each infinite Boolean algebra (BA), and the behaviour of such functions under ultraproducts. For some common functions k we havefor others we have ≤ instead, under suitable assumptions. For the function π character we go into more detail. More specifically, ≥ holds when F is regular, for cellularity, length, irredundance, spread, and incomparability. ≤ holds for π. ≥ holds under GCH for F regular, for depth, π, πχ, χ, h-cof, tightness, hL, and hd. These results show that ≥ can consistently hold in ZFC since if V = L holds then all uniform ultrafilters are regular. For π-character we prove two more results: (1) If F is regular and ess , then(2) It is relatively consistent to have , where A is the denumerable atomless BA.A thorough analysis of what happens without the assumption that F is regular can be found in Rosłanowski, Shelah [8] and Magidor, Shelah [5]. Those papers also mention open problems concerning the above two possible inequalities.

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