baire category theorem
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Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3567-3580 ◽  
Author(s):  
Alexander Sostak

An important class of spaces was introduced by I.A. Bakhtin (under the name ?metric-type?) and independently rediscovered by S. Czerwik (under the name ?b-metric?). Metric-type spaces generalize ?classic? metric spaces by replacing the triangularity axiom with a more general axiom d(x,z)? k? (d(x,y)+ d(y,z)) for all x,y,z ? X where k ? 1 is a fixed constant. Recently R. Saadadi has introduced the fuzzy version of ?metric-type? spaces. In this paper we consider topological and sequential properties of such spaces, illustrate them by several examples and prove a certain version of the Baire Category Theorem.


2018 ◽  
Vol 59 (4) ◽  
pp. 605-636 ◽  
Author(s):  
Vasco Brattka ◽  
Matthew Hendtlass ◽  
Alexander P. Kreuzer

2016 ◽  
Vol 37 (4) ◽  
pp. 1308-1322 ◽  
Author(s):  
JIMMY TSENG

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of $C^{2}$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.


2014 ◽  
Vol 25 (7) ◽  
pp. 1607-1625 ◽  
Author(s):  
REINHOLD HECKMANN

The first part of the paper presents a generalization of the well-known Baire category theorem. The generalization consists in replacing the dense open sets of the original formulation by dense UCO sets, where UCO means union of closed and open. This topological theorem is exactly what is needed to prove in the second part of the paper the locale-theoretic result that locales whose frame of opens has a countable presentation (countably many generators and countably many relations) are spatial. This spatiality theorem does not require choice.


2014 ◽  
Vol 226 (2) ◽  
pp. 123-130
Author(s):  
Jonathan Brown ◽  
Lisa Orloff Clark

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