COUNTABLE SPACES WITH PEANO PROPERTY

In 1890, Giuseppe Peano  published an example of a continuous curve passing through every point of the square $[0,1]^2$. A curve with such properties is called a Peano curve. In fact, Peano constructed a continuous surjective mapping from the unit segment $[0,1]$ to the square $[0,1]^2$. Peano's research was motivated by one   result of George Cantor that the set of points of a unit segment has the same cardinality as the set of points of a unit square.In 1890, Giuseppe Peano  published an example of a continuous curve passing through every point of the square $[0,1]^2$. A curve with such properties is called a Peano curve. In fact, Peano constructed a continuous surjective mapping from the unit segment $[0,1]$ to the square $[0,1]^2$. Peano's research was motivated by one   result of George Cantor that the set of points of a unit segment has the same cardinality as the set of points of a unit square. According to the Hahn-Mazurkevich theorem the Hausdorff topological space $X$ is a continuous image of a unit segment $[0,1]$ if and only if when $X$ is compact, metrizable, connected, locally connected and nonempty. The Hausdorff continuous image of a segment is called {\it Peano space} or {\it Peano continuum}. Sierpinski proved that a connected compact metric space $X$ is a Peano continuum if and only if for every $\varepsilon>0$ the space $X$ can be covered by connected sets of the diameter $\le\varepsilon$. Therefore, naturally arises question about the investigation of disconnected metric spaces $X$ for which there is a continuous surjection between $X$ and $X^2$. Sierpinski   characterized rational numbers as a metric countable space without isolated points. Hausdorff  described irrational numbers as a metric, separable, completely metrizable, zero-dimensional and nowhere locally compact space. It follows, in particular, that the square $\mathbb Q^2$ is a continuous image of the set $\mathbb Q$ and the square of irrational numbers is a continuous image of the set   of irrational numbers. Thus, it would be interesting to find a description of other disconnected subsets of the real line, except those that are homeomorphic to $\mathbb Q$ or $\mathbb R\setminus Q$. In this article we will focus on countable sets such that the set of isolated points of which may not be empty. The main result is the following (see Theorem 2): the square of a countable regular topological space $X$ is its continuous image if and only if $X$ is not compact.