Is $U_a=d^{-1}([0,a])= \{(m,n) \in M \times M: d(m,n) \leq a\}$ open set in a metric space $M \times M$ for positive real $a$?
My attempt: I think it is not an open set. Because let $M=\mathbb{R}$ and $d$ is a usual (distant) metric. Consider $a=1$, then there is no open ball around point $(1,2)$, which is contained in $U_a$.
However, recently when I read a research paper, it is written that it is an open set. So I got confused.
The link to the research paper is here. In the paper, on the page number 151, read example 12.
The paper is: Das, T., Lee, K., Richeson, D. S., & Wiseman, J. (2013). Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces. Topology and Its Applications, 160(1), 149–158. https://doi.org/10.1016/j.topol.2012.10.010
Please help. Thanks.
