# Net (of sets in a topological space)

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

network (of sets in a topological space)

A family ${\mathcal P}$ of subsets of a topological space $X$ such that for each $x \in X$ and each neighbourhood $O _ {x}$ of $x$ there is an element $M$ of ${\mathcal P}$ such that $x \in M \subset O _ {x}$.

The family of all one-point subsets of a space and every base of a space are networks. The difference between a network and a base is that the elements of a network need not be open sets. Networks appear under continuous mappings: If $f$ is a continuous mapping of a topological space $X$ onto a topological space $Y$ and ${\mathcal B}$ is a base of $X$, then the images of the elements of ${\mathcal B}$ under $f$ form a network ${\mathcal P} = \{ {f U } : {U \in {\mathcal B} } \}$ in $Y$. Further, if $X$ is covered by a family $\{ {X _ \alpha } : {\alpha \in A } \}$ of subspaces, then, taking for each $\alpha \in A$ any base ${\mathcal B} _ \alpha$ of $X _ \alpha$ and amalgamating these bases, a network ${\mathcal P} = \cup \{ { {\mathcal B} _ \alpha } : {\alpha \in A } \}$ in $X$ is obtained. Spaces with a countable network are characterized as images of separable metric spaces under continuous mappings.

The minimum cardinality of a network of a space $X$ is called the network weight, or net weight, of $X$ and is denoted by $\mathop{\rm nw} ( X)$. The net weight of a space never exceeds its weight (cf. Weight of a topological space), but, as is shown by the example of a countable space without a countable base, the net weight can differ from the weight. For compact Hausdorff spaces the net weight coincides with the weight. This result extends to locally compact spaces, Čech-complete spaces and feathered spaces (cf. Feathered space). Hence, in particular, it follows that weight does not increase under surjective mappings of such spaces. Another corollary: If a feathered space $X$( in particular, a Hausdorff compactum) is given as the union of a family of cardinality $\leq \tau$ of subspaces, the weight of each of which does not exceed $\tau$, supposed infinite, then the weight of $X$ does not exceed $\tau$.

#### References

 [1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) [2] A.V. Arkhangel'skii, "An addition theorem for weights of sets lying in bicompacta" Dokl. Akad. Nauk SSSR , 126 : 2 (1959) pp. 239–241 (In Russian)

A net in a set (topological space) $X$ is an indexed set $\{ x _ \alpha \} _ {\alpha \in \Sigma }$ of points of $X$, where $\Sigma$ is a directed set. In Russian this is called a generalized sequence.