MATHEMATICS

On modification of the Sorgenfrey line

E. S. Sukhacheva, T. E. Khmyleva

Tomsk State University, Russian Federation

Abstract: In this paper, we consider a topological space $S_A$ that is a modification of the Sorgenfrey line $S$ and is defined as follows: if a point $x\in A\subset \mathbf{R}$, then the base of neighborhoods of the point is $\{[x, x+\varepsilon), \forall\varepsilon>0\}$; if a point $x\in \mathbf{R}\setminus A$, then the base of neighborhoods of the point is $\{(x-\varepsilon, x], \forall\varepsilon>0\}$. The following criterion for a homeomorphism of the spaces $S_A$ and $S_Q$ has been obtained: the spaces $S_A$ and $S_Q$ are homeomorphic if and only if a subset $A\subset S_A$ is countable and dense in $S$.

Keywords: Sorgenfrey line, homeomorphism, Baire space, the space of the second category.

UDC: 515.12

Received: 10.02.2017

DOI: 10.17223/19988621/46/5

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