Sum Theorems for Countably Paracompact Spaces

1973 ◽  
Vol 25 (4) ◽  
pp. 706-711
Author(s):  
Henry Potoczny
Keyword(s):  
Open Cover ◽  
Equivalent Formulation ◽  
Countably Paracompact ◽  
Definition Of ◽  
Paracompact Spaces ◽  
Countable Paracompactness

In this paper, we extend the class of spaces to which the Σ and β theorems of Hodel apply, as well as the sum and subset theorems of [2]. Instead of the open cover definition of countable paracompactness, we utilize an equivalent formulation of countable paracompactness, due to Ishikawa [3].

scholarly journals Characterizations of Strongly Paracompact Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Xin Zhang
Keyword(s):  
Open Cover ◽  
Normal Space ◽  
Compact Spaces ◽  
Countably Paracompact ◽  
Star Countable ◽  
Perfectly Normal Space ◽  
Paracompact Spaces ◽  
Perfectly Normal

Characterizations of strongly compact spaces are given based on the existence of a star-countable open refinement for every increasing open cover. It is proved that a countably paracompact normal space (a perfectly normal space or a monotonically normal space) is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement. Moreover, it is shown that a space is linearlyDprovided that every increasing open cover of the space has a point-countable open refinement.

scholarly journals Paracompactness with respect to an ideal

1997 ◽  
Vol 20 (3) ◽  
pp. 433-442 ◽  
Author(s):  
T. R. Hamlett ◽  
David Rose ◽  
Dragan Janković
Keyword(s):  
Topological Space ◽  
Finite Union ◽  
Open Cover ◽  
Locally Finite ◽  
Open Set ◽  
Special Cases ◽  
Paracompact Spaces

An ideal on a setXis a nonempty collection of subsets ofXclosed under the operations of subset and finite union. Given a topological spaceXand an idealℐof subsets ofX,Xis defined to beℐ-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all ofXexcept for a set inℐ. Basic results are investigated, particularly with regard to theℐ- paracompactness of two associated topologies generated by sets of the formU−IwhereUis open andI∈ℐand⋃{U|Uis open andU−A∈ℐ, for some open setA}. Preservation ofℐ-paracompactness by functions, subsets, and products is investigated. Important special cases ofℐ-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of Singal and Arya [“On m-paracompact spaces”, Math. Ann., 181 (1969), 119-133].

On Countably Paracompact Spaces

1951 ◽  
Vol 3 ◽  
pp. 219-224 ◽  
Author(s):  
C. H. Dowker
Keyword(s):  
Topological Space ◽  
Finite Number ◽  
Finite Collection ◽  
Countable Collection ◽  
Locally Finite ◽  
Open Set ◽  
Countably Paracompact ◽  
Paracompact Spaces ◽  
Countable Covering

Let X be a topological space, that is, a space with open sets such that the union of any collection of open sets is open and the intersection of any finite number of open sets is open. A covering of X is a collection of open sets whose union is X. The covering is called countable if it consists of a countable collection of open sets or finite if it consists of a finite collection of open sets ; it is called locally finite if every point of X is contained in some open set which meets only a finite number of sets of the covering. A covering is called a refinement of a covering U if every open set of X is contained in some open set of . The space X is called countably paracompact if every countable covering has a locally finite refinement.

scholarly journals A new bitopological paracompactness

1986 ◽  
Vol 41 (2) ◽  
pp. 268-274 ◽  
Author(s):  
T. G. Raghavan ◽  
I. L. Reilly
Keyword(s):  
Open Cover ◽  
Locally Finite ◽  
Quasimetric Space ◽  
Paracompact Spaces ◽  
Bitopological Spaces

AbstractIn this paper we define generalization of paracompactness for bitopological spaces. (X, τ1, τ2) is Δ-pairwise paracompact if and only if every τi open cover admits a τ1 ∨ τ2 open refinement which is τ1 ∨ τ2 locally finite. Every quasimetric space (X, τp, τq) is Δ-pairwise paracompact. An analogue of Michael's characterization of regular paracompact spaces is proved for Δ-pairwise paracompact spaces.

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