<abstract xmlns="http://www.w3.org/1999/xhtml"> Let <italic>C</italic>0 denote the set of all non-decreasing continuous functions <italic>f </italic>: (0<italic>, </italic>1] →<italic></italic>(0<italic>, </italic>1] such that lim<italic>x</italic>→0+ ƒ(<italic>x</italic>) = 0 and ƒ(<italic>x</italic>) <italic>≤ x </italic>for <italic>x </italic>∈(0<italic>, </italic>1] and let <italic>A </italic>be a measurable subset of the plane. We define the notion of a density point of <italic>A </italic>with respect to ƒ. This is a starting point to introduce the mapping <italic>D</italic>ƒ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping <italic>D</italic>ƒ leads to the topology <italic>T</italic>ƒ, analogously as for the density topology. The properties of the topologies <italic>T</italic>ƒ are considered.</abstract>

Let C0 denote the set of all non-decreasing continuous functions f : (0, 1] →(0, 1] such that limx→0+ ƒ(x) = 0 and ƒ(x) ≤ x for x ∈(0, 1] and let A be a measurable subset of the plane. We define the notion of a density point of A with respect to ƒ. This is a starting point to introduce the mapping Dƒ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping Dƒ leads to the topology Tƒ, analogously as for the density topology. The properties of the topologies Tƒ are considered.

Density topologies on the plane between ordinary and strong

Faculty of Mathematics and Computer Science Banacha 22 PL–90-238 Łódž POLAND

The College of Computer Science Chair of Mathematics and Physics Rzgowska  17A PL–93-008 Łódž POLAND

Tatra Mountains Mathematical Publications

{"article-title":"Density topologies on the plane between ordinary and strong"}

Let C0 denote the set of all non-decreasing continuous functions f : (0, 1] →(0, 1] such that limx→0+ ƒ(x) = 0 and ƒ(x) ≤ x for x ∈(0, 1] and let A be a...

Density topologies on the plane between ordinary and strong

Publié en ligne: 12 nov. 2012

Pages: 139 - 151

DOI: https://doi.org/10.2478/v10127-009-0054-1

Mots clés
density point, density topology, density point with respect to ƒ.

This content is open access.

Density topologies on the plane between ordinary and strong

Publié en ligne: 12 nov. 2012

Pages: 139 - 151

DOI: https://doi.org/10.2478/v10127-009-0054-1

Mots clésdensity point, density topology, density point with respect to ƒ.

This content is open access.

Mots clés
density point, density topology, density point with respect to ƒ.