scholarly journals Test map characterizations of local properties of fundamental groups

2018 ◽  
Vol 12 (01) ◽  
pp. 37-85 ◽  
Author(s):  
Jeremy Brazas ◽  
Hanspeter Fischer
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closure Operator ◽  
Fundamental Groups ◽  
Space Theory ◽  
Unified Approach ◽  
Covering Map ◽  
Unique Path ◽  
Local Properties ◽  
Path Connected

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any path-connected metric space [Formula: see text], and a subgroup [Formula: see text], we characterize the unique path lifting property relative to [Formula: see text] in terms of a new closure operator on the [Formula: see text]-subgroup lattice that is induced by maps from a fixed “test” domain into [Formula: see text]. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

scholarly journals On locally 1-connectedness of quotient spaces and its applications to fundamental groups

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 27-35
Author(s):  
Ali Pakdaman ◽  
Hamid Torabi ◽  
Behrooz Mashayekhy
Keyword(s):  
Fundamental Group ◽  
Metric Space ◽  
Pairwise Disjoint ◽  
Quotient Space ◽  
Continuous Homomorphism ◽  
Fundamental Groups ◽  
Quotient Spaces ◽  
Path Connected

Let X be a locally 1-connected metric space and A1,A2,...,An be connected, locally path connected and compact pairwise disjoint subspaces of X. In this paper, we show that the quotient space X/(A1,A2,..., An) obtained from X by collapsing each of the sets Ai?s to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space X/(A1,A2,...,An).

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

scholarly journals Equivalent conditions for digital covering maps

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4005-4014
Author(s):  
Ali Pakdaman ◽  
Mehdi Zakki
Keyword(s):  
Local Isomorphism ◽  
Covering Map ◽  
Unique Path ◽  
Continuous Surjection ◽  
Digital Covering ◽  
Equivalent Conditions ◽  
Digital Spaces

It is known that every digital covering map p:(E,k) ? (B,?) has the unique path lifting property. In this paper, we show that its inverse is true when the continuous surjective map p has no conciliator point. Also, we prove that a digital (k,?)-continuous surjection p:(E,k)? (B,?) is a digital covering map if and only if it is a local isomorphism, when all digital spaces are connected. Moreover, we find out a loop criterion for a digital covering map to be a radius n covering map.

More on Extending Continuous Pseudometrics

1970 ◽  
Vol 22 (5) ◽  
pp. 984-993 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closed Subset ◽  
Locally Finite ◽  
Whole Space

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

scholarly journals Fuzzyθ-closure operator on fuzzy topological spaces

1991 ◽  
Vol 14 (2) ◽  
pp. 309-314 ◽  
Author(s):  
M. N. Mukherjee ◽  
S. P. Sinha
Keyword(s):  
Topological Space ◽  
Fuzzy Sets ◽  
Closure Operator ◽  
Topological Spaces ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The paper contains a study of fuzzyθ-closure operator,θ-closures of fuzzy sets in a fuzzy topological space are characterized and some of their properties along with their relation with fuzzyδ-closures are investigated. As applications of these concepts, certain functions as well as some spaces satisfying certain fuzzy separation axioms are characterized in terms of fuzzyθ-closures andδ-closures.

Variable Neighborhood Model for Agent Control Introducing Accessibility Relations Between Agents with Linear Temporal Logic

2014 ◽  
Vol 18 (6) ◽  
pp. 937-945
Author(s):  
Seiki Ubukata ◽  
◽  
Tetsuya Murai ◽  
Yasuo Kudo ◽  
Seiki Akama ◽  
...  
Keyword(s):  
Topological Space ◽  
Temporal Logic ◽  
Binary Relation ◽  
Linear Temporal Logic ◽  
Binary Relations ◽  
Cellular Space ◽  
Current Time ◽  
Low Level ◽  
Local Properties ◽  
Agent Control

In general, there are two types of agents, reflex and deliberative. The former does not have the ability for deep planning that produces higher-level actions to attain goals cooperatively, which is the ability of the latter. Can we cause reflex agents to act as though they could plan their actions? In this paper, we propose a variable neighborhood model for reflex agent control, that allows such agents to create plans in order to attain their goals. The model consists of three layers: (1) topological space, (2) agent space, and (3) linear temporal logic. Agents with their neighborhoods move in a topological space, such as a plane, and in a cellular space. Then, a binary relation between agents is generated each time from the agents’ position and neighborhood. We call the pair composed of a set of agents and binary relations the agent space. In order to cause reflex agents to have the ability to attain goals superficially, we consider the local properties of the binary relation between agents. For example, if two agents have a symmetrical relation at the current time, they can struggle to maintain symmetry or they could abandon symmetry at the next time, depending on the context. Then, low-level behavior, that is, the maintenance or abandonment of the local properties of binary relations, grant reflex agents a method for selecting neighborhoods for the next time. As a result, such a sequence of low-level behavior generates seemingly higher-level actions, as though reflex agents could attain a goal with such actions. This low-level behavior is shown through simulation to generate the achievement of a given goal, such as cooperation and target pursuing.

A theorem about countable decomposability

1982 ◽  
Vol 91 (3) ◽  
pp. 457-458 ◽  
Author(s):  
Roy O. Davies ◽  
Claude Tricot
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Continuous Functions ◽  
Perfect Set ◽  
Baire Function ◽  
Separable Metric Space

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

scholarly journals A Metric Space Approach on the Molecular vs. Chemical Similarity of Some Analgesic and Euphoric Compounds

2019 ◽  
Author(s):  
Demetrios Xenides ◽  
Dionisia Fostiropoulou ◽  
Dimitrios S Vlachos
Keyword(s):  
Metric Space ◽  
Clinical Findings ◽  
Space Theory ◽  
Chemical Similarity ◽  
Analgesic Drugs ◽  
Structural Indices ◽  
Space Approach

<p>There is a relentless effort on gaining information on the reason why some compounds could cause similar effects though they are or not structural similar. That is the chemical similarity that plays an equally important role and we approach it via metric space theory on a set of analgesic drugs and euphoric compounds. The findings of the present study are in agreement to these obtained via traditional structural indices moreover are in accord with clinical findings.</p>

scholarly journals Bombay hypertopologies

2003 ◽  
Vol 4 (2) ◽  
pp. 421 ◽  
Author(s):  
Giuseppe Di Maio ◽  
Enrico Meccariello ◽  
Somashekhar Naimpally
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Wijsman Convergence ◽  
Locally Finite ◽  
Simple Result ◽  
Special Cases ◽  
Wijsman Topology

<p>Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.</p>

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