Even Covers and Collectionwise Normal Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 466-473 ◽  
Author(s):  
H. L. Shapiro ◽  
F. A. Smith
Keyword(s):  
Sufficient Conditions ◽  
Open Cover ◽  
Locally Finite ◽  
Finite Open Cover ◽  
Elementary Topology

The concept of an even cover is introduced early in elementary topology courses and is known to be valuable. Among other facts it is known that X is paracompact if and only if every open cover of X is even. In this paper we introduce the concept of an n-even cover and show its usefulness. Using n-even we define an embedding that on closed subsets is equivalent to collectionwise normal. We also give sufficient conditions for a point finite open cover to have a locally finite refinement and also sufficient conditions for this refinement to be even. Finally we show that the collection of all neighborhoods of the diagonal of X is a uniformity if and only if every even cover is normal. This last result is particularly interesting in light of the fact that every normal open cover is even.

scholarly journals The Structure of Locally Finite Two-Connected Graphs

10.37236/1211 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Carl Droms ◽  
Brigitte Servatius ◽  
Herman Servatius
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Connected Graphs ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Necessary And Sufficient ◽  
Block Tree ◽  
Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

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].

scholarly journals Fecundity regulation in a spatial birth-and-death process

2020 ◽  
Vol 21 (01) ◽  
pp. 2050038 ◽  
Author(s):  
Viktor Bezborodov ◽  
Luca Di Persio ◽  
Dmitri Finkelshtein ◽  
Yuri Kondratiev ◽  
Oleksandr Kutoviy
Keyword(s):  
Level Sets ◽  
Existence And Uniqueness ◽  
Ecological Model ◽  
Sufficient Conditions ◽  
Death Process ◽  
Regulation Mechanism ◽  
Birth And Death Process ◽  
Locally Finite ◽  
Return Times ◽  
Time Space

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density-dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster–Lyapunov-type function, we study return times to certain level sets of tempered configurations. We also find sufficient conditions that the degenerate invariant distribution is unique for the considered process.

Sufficient conditions for expansive group action

2019 ◽  
Vol 20 (03) ◽  
pp. 2050022 ◽  
Author(s):  
Ali Barzanouni
Keyword(s):  
Solvable Group ◽  
Group Action ◽  
Uniform Space ◽  
Sufficient Conditions ◽  
Open Cover ◽  
Probability Measures ◽  
Continuous Action ◽  
Paracompact Space ◽  
Finite Set ◽  
Borel Probability

Existence of expansivity for group action [Formula: see text] depends on algebraic properties of [Formula: see text] and the topology of [Formula: see text]. We give an expansive action of a solvable group on [Formula: see text] while there is no expansive action of a solvable group on a dendrite [Formula: see text]. We prove that a continuous action [Formula: see text] on a compact metric space [Formula: see text] is expansive if and only if there exists an open cover [Formula: see text] such that for any other open cover [Formula: see text], [Formula: see text] for some finite set [Formula: see text]. In this paper, we introduce the notion of topological expansivity of a group action [Formula: see text] on a [Formula: see text]-paracompact space [Formula: see text]. If a [Formula: see text]-paracompact space [Formula: see text] admits topologically expansive action, then [Formula: see text] is Hausdorff space. We also show that a continuous action [Formula: see text] of a finitely generated group [Formula: see text] on a compact Hausdorff uniform space [Formula: see text] is expansive with an expansive neighborhood [Formula: see text] if and only if for every [Formula: see text] there is an entourage [Formula: see text] such that for every two [Formula: see text]-pseudo orbit [Formula: see text] if [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text]. Finally, we introduce measure [Formula: see text]-expansive actions on a uniform space. The set of all [Formula: see text]-expansive measures with common expansive neighborhood for a group action [Formula: see text] is a convex, closed and [Formula: see text]-invariant subset of the set of all Borel probability measures on [Formula: see text]. Also, we show that a group action [Formula: see text] is expansive if all Borel probability measures are [Formula: see text]-expansive or all Dirac measures [Formula: see text], [Formula: see text], have a common expansive neighborhood.

Matchings in Arbitrary Graphs

1971 ◽  
Vol 69 (3) ◽  
pp. 401-407 ◽  
Author(s):  
R. A. Brualdi
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Maximum Cardinality ◽  
Original Proof ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

A variation on the variational principle and applications to entropy pairs

1997 ◽  
Vol 17 (1) ◽  
pp. 29-43 ◽  
Author(s):  
F. BLANCHARD ◽  
E. GLASNER ◽  
B. HOST
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Open Cover ◽  
Topological Dynamical System ◽  
Positive Topological Entropy ◽  
Finite Open Cover

The variational principle states that the topological entropy of a topological dynamical system is equal to the sup of the entropies of invariant measures. It is proved that for any finite open cover there is an invariant measure such that the topological entropy of this cover is less than or equal to the entropies of all finer partitions. One consequence of this result is that for any dynamical system with positive topological entropy there exists an invariant measure whose set of entropy pairs is equal to the set of topological entropy pairs.

scholarly journals On generalization of decomposability

1981 ◽  
Vol 22 (1) ◽  
pp. 77-81 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Bounded Linear Operator ◽  
Invariant Subspaces ◽  
Complex Banach Space ◽  
Open Cover ◽  
Bounded Linear ◽  
Image Position ◽  
Finite Open Cover

Let X be a complex Banach space and let T be a bounded linear operator on X. Then T is decomposable if for every finite open cover of σ(T) there are invariant subspaces Yi(i= 1, 2, …, n) such that(An invariant subspace Y is spectral maximal [for T] if it contains every invariant subspace Z for which σ(T|Z) ⊂ σ(T|Y).).

scholarly journals On paracompact regular spaces

1961 ◽  
Vol 2 (2) ◽  
pp. 147-150
Author(s):  
Shuen Yuan
Keyword(s):  
Topological Space ◽  
Finite Number ◽  
Recent Result ◽  
Open Cover ◽  
Regular Space ◽  
Locally Finite ◽  
Normality Condition ◽  
Discrete Family

A topological space is paracompact if and only if each open cover of the space has an open locally finite refinement. It is well-known that an unusual normality condition is satisfied by each paracompact regular space X [p. 158, 5]: Let α be a locally finite (discrete) family of subsets of X, then there is a neighborhood V of the diagonal Δ(X) (in X × X), such that V[x] intersects at most a finite number of members (respectively at most one member) of {V[A]: A ∈ α} for each x ∈ X. In this not we will show that a variant of this condition actually characterizes paracompactness. Among other results, an improvement to a recent result of H. H.Corson [2] is given so as to accord with a conjecture of J. L. Kelley [p. 208, 5] more prettily, and we connect paracompactness to metacompactness [1]

scholarly journals Generalized products of weakly m — n compact spaces, I

1982 ◽  
Vol 33 (2) ◽  
pp. 143-151 ◽  
Author(s):  
U. N. B. Dissanayake
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Open Cover ◽  
Compact Spaces

AbstractA topological space X is said to be weakly-Lindelöf if and only if every open cover of X has a countable sub-family with dense union. We know that products of two Lindelöf spaces need not be weakly-Lindelöf. In this paper we obtain non-trivial sufficient conditions on small sub-products to ensure the producitivity of the property weakly-Lindelöf with respect to arbitrary products.

Matchings in Countable Graphs

1977 ◽  
Vol 29 (1) ◽  
pp. 165-168 ◽  
Author(s):  
K. Steffens
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

Tutte [9] has given necessary and sufficient conditions for a finite graph to have a perfect matching. Different proofs are given by Brualdi [1] and Gallai [2; 3]. The shortest proof of Tutte's theorem is due to Lovasz [5]. In another paper [10] Tutte extended his conditions for a perfect matching to locally finite graphs. In [4] Kaluza gave a condition on arbitrary graphs which is entirely different from Tutte's.

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