On local weak $\tau$-density of topological spaces

Questions of local weak $\tau$-density of topological spaces are studied. Sufficient conditions are found for the preservation of the property of local weak $\tau$-density of subsets of topological spaces. It is shown that a subset of a locally dense space is also locally weakly $\tau$-dense if it satisfies at least one of the following conditions: a) the subset is open in space; b) the subset is everywhere dense in space; c) the subset is canonically closed in space.