Linear hyperplane arrangements and local finiteness

PDF p. 4: §2.1

Let $E$ be a real vector space equipped with a specified topology, and give each subset of $E$ the subspace topology: its open sets are its intersections with open subsets of $E$. In the finite-dimensional setting one may use the Euclidean topology described in the arrangement interface. A subset $T\subseteq E$ is convex when $(1-t)x+ty\in T$ for all $x,y\in T$ and $0\le t\le 1$. A linear hyperplane is a linear subspace $H\subseteq E$ of codimension one, meaning that $E\mathrm{/}H$ has dimension one.

A hyperplane arrangement $(T,\mathcal{H})$ consists of a convex subset $T\subseteq E$ and a set $\mathcal{H}$ of linear hyperplanes in $E$, possibly infinite, such that $H\cap T\mathrm{\ne }\mathrm{\varnothing }$ for every $H\in \mathcal{H}$.

For a convex subset $T^{\mathrm{\prime }}\subseteq T$, the restriction to $T^{\mathrm{\prime }}$ is the arrangement $(T^{\mathrm{\prime }},{\mathcal{H}}_{T^{\mathrm{\prime }}})$, where ${\mathcal{H}}_{T^{\mathrm{\prime }}}=\{H\in \mathcal{H}:H\cap T^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }\}$. It is also denoted $(T^{\mathrm{\prime }},\mathcal{H})$ when the restriction of the hyperplane set is understood.

The arrangement is locally finite if, for every $x\in T$, there is an open set $U$ in the subspace topology on $T$ with $x\in U$ such that $U$ meets only finitely many distinct members of $\mathcal{H}$.

Vector space over a field

The quotient vector space V/W and its canonical projection

Locally finite arrangements and relative face terminology

Linear hyperplane

Linear subspace of a vector space

Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis

Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

The specified topology gives a subspace topology on $T$, so the neighbourhood condition is meaningful. Convexity of $T^{\mathrm{\prime }}$ and the defining filter on ${\mathcal{H}}_{T^{\mathrm{\prime }}}$ ensure that the restriction again satisfies the arrangement definition. The members retained are hyperplanes in $E$ themselves, so distinct hyperplanes are counted separately even if their intersections with $T$ coincide. The finite-dimensional convention and these arrangement operations agree with the arrangement interface.

The cited article supplies the arrangement terminology and its restriction and local-finiteness conventions. The explicit choice of topology makes the neighbourhood convention precise; no particular topology is asserted for an arbitrary-dimensional $E$.