arXiv subjects

Andrés Merino

Publications and source records attributed to Andrés Merino.

3 recordsLinked to original sources

arXiv2020

Some properties related to the Cantor-Bendixson derivative on a Polish space

We show a necessary and sufficient condition for any ordinal number to be a Polish space. We also prove that for each countable Polish space, there exists a countable ordinal number that is an upper bound for the first component of the Cantor-Bendixson characteristic of every compact countable subset of the aforementioned space. In addition, for any uncountable Polish space, for every countable ordinal number and for all nonzero natural number, we show the existence of a compact countable subset of this space such that its Cantor-Bendixson characteristic equals the previous pair of numbers. Finally, for each Polish space, we determine the cardinality of the partition, up to homeomorphisms, of the set of all compact countable subsets of the aforesaid space.

math.GM↗

arXiv2018

Countable ordinal spaces and compact countable subsets of a metric space

We show in detail that every compact countable subset of a metric space is homeomorphic to a countable ordinal number, which extends a result given by Mazurkiewicz and Sierpinski for finite-dimensional Euclidean spaces. In order to achieve this goal, we use Transfinite Induction to construct a specific homeomorphism. In addition, we prove that for all metric space $(E,d)$, the cardinality of the set of all the equivalence classes $\mathscr{K}_E$, up to homeomorphisms, of compact countable subsets of $E$ is less than or equal to $\aleph_1$, i.e. $|\mathscr{K}_E| \le \aleph_1$. We also show that for all cardinal number $κ$ smaller than or equal to $\aleph_1$, there exists a metric space $(E_κ, d_κ)$ such that $|\mathscr{K}_{E_κ}|= κ$.

math.GN↗

arXiv2016

A primitive associated to the Cantor-Bendixson derivative on the real line

We consider the class of compact countable subsets of the real numbers $\mathbb{R}$. By using an appropriate partition, up to homeomorphism, of this class we give a detailed proof of a result shown by S. Mazurkiewicz and W. Sierpinski related to the cardinality of this partition. Furthermore, for any compact subset of $\mathbb{R}$, we show the existence of a "primitive" related to its Cantor-Bendixson derivative.

math.GN↗