Local $\tau$-density and local weak $\tau$-density in topological spaces

The article considers locally compact, metrizable, and linearly ordered spaces. In these spaces, the concepts of local $\tau$-density and local weak $\tau$-density are considered. It is shown that in locally compact, metrizable, and linearly ordered spaces the concepts of local $\tau$-density and local weak $\tau$-density coincide. It is proved that a locally weakly $\tau$-dense subset of a compact set with the first axiom of countability is locally $\tau$-dense.