Locally finite arrangements and relative face terminology

Let $E$ be a finite-dimensional real vector space, let $n={\dim }_{\mathbb{R}}E$, and choose a linear coordinate isomorphism $c:E\to {\mathbb{R}}^n$. On ${\mathbb{R}}^n$ use the Euclidean metric $d_2(u,v)=({\sum }_{i=0}^{n-1}(u_i-v_i{)}^2{)}^{1\mathrm{/}2}$. Give $E$ the metric $d_E(x,y)=d_2(c(x),c(y))$ and its metric topology: $O\subseteq E$ is open if for each $x\in O$ there is $r>0$ such that $\{y\in E:d_E(x,y)<r\}\subseteq O$. For $n=0$, ${\mathbb{R}}^0$ and $E$ are singletons, the sum is empty and the metric is zero; the topology is $\{\mathrm{\varnothing },E\}$. This specifies the transported Euclidean topology. A subset $T\subseteq E$ is convex if $(1-t)x+ty\in T$ whenever $x,y\in T$ and $0\le t\le 1$. Give $T$ the subspace topology: its open sets are the intersections of $T$ with open subsets of $E$. A neighbourhood of $x\in T$ is a subset of $T$ containing an open subset of $T$ that contains $x$.

An arrangement $(T,\mathcal{A})$ consists of such a convex $T$ and a set $\mathcal{A}$ of linear hyperplanes $H\subseteq E$, each meeting $T$. A linear hyperplane is a linear subspace of codimension one. The arrangement is locally finite in $T$ if every $x\in T$ has a neighbourhood meeting only finitely many distinct members of $\mathcal{A}$. For a convex $T^{\mathrm{\prime }}\subseteq T$, its restriction has hyperplane set $\{H\in \mathcal{A}:H\cap T^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }\}$.

A chamber is a connected component of $T\setminus {\bigcup }_{H\in \mathcal{A}}H$, with the subspace topology. For a chamber $C$, write $\bar{C}$ for its closure in $E$: the points whose every open neighbourhood in $E$ meets $C$. An affine subspace of $E$ is a translate $a+U$ of a linear subspace $U\subseteq E$. Its direction space is $U=(a+U)-a$, and its affine dimension is ${\dim }_{\mathbb{R}}U$. For a nonempty set $A\subseteq E$, its affine hull $\mathrm{aff}(A)$ is the intersection of all affine subspaces containing $A$; its affine dimension is the affine dimension of $\mathrm{aff}(A)$. A wall of $C$ is a nonempty set $\bar{C}\cap H$ of affine dimension ${\dim }_{\mathbb{R}}E-1$, where $H\in \mathcal{A}$. No affine dimension is assigned to an empty intersection here. The wall is the intersection, while $H$ is its defining arrangement hyperplane; these are different kinds of objects. Two distinct chambers are adjacent if they share a wall. A gallery is a finite sequence of chambers in which consecutive chambers are adjacent; its length is the number of consecutive pairs.

A basis of a vector space is a linearly independent spanning set. An open simplicial cone in a finite-dimensional real vector space $F$ is a set $\{{\sum }_{i=1}^k{t}_i{v}_i:t_i>0\}$ for a basis $v_1,\dots ,v_k$ of $F$; for $k=0$ the empty sum is $0$. For an arrangement with $T=E$, put $L={\bigcap }_{H\in \mathcal{A}}H$, taking $L=E$ when $\mathcal{A}$ is empty. The quotient vector space $E\mathrm{/}L$ consists of cosets $x+L$ with operations $(x+L)+(y+L)=(x+y)+L$ and $t(x+L)=tx+L$; its canonical projection is $q:E\to E\mathrm{/}L$, $q(x)=x+L$. The arrangement is simplicial if every chamber is of the form $L+\{{\sum }_{i=1}^k{t}_i{v}_i:t_i>0\}$, where $q(v_1),\dots ,q(v_k)$ form a basis of $E\mathrm{/}L$. Thus the common linear factor is retained in $E$ and removed for the simplicial test.

Vector space over a field

Linear subspace of a vector space

Invertible linear maps, linear isomorphisms, and inverse linear maps

Linear hyperplane

A convex subset of ℝᵐ contains every line segment between two of its points

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

Connected components, quasicomponents, and totally disconnected spaces

Refinements, locally finite families, point-finite families, and star refinements

Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis

Affine subspaces as translates x+U of linear subspaces

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

The quotient vector space V/W and its canonical projection

ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it

The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement

Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric

If dim_F V = n and U is a linear subspace of V, then U is finite-dimensional, dim_F U ≤ n, and dim_F U = n if and only if U = V

For $n\ge 1$, the Euclidean metric supplier establishes that $d_2$ is a metric; transporting its distances along the bijection $c$ gives the metric axioms for $d_E$. For $n=0$, the zero metric on the singleton satisfies these axioms directly. Thus the metric topology and the coordinate isomorphism specify the topology used in all closure and neighbourhood statements. The subspace topology makes the stated relative neighbourhoods meaningful. Local finiteness applies the usual neighbourhood condition to the hyperplanes indexed by the distinct members of $\mathcal{A}$; it does not identify different hyperplanes merely because their traces on $T$ coincide. Connected components supply the chambers. The direction space of an affine subspace is independent of its chosen point: if $b\in a+U$, then $(a+U)-b=U$. For a nonempty $A$, fix $a\in A$; every affine subspace containing $A$ is $a+U$, so their intersection is $a+\bigcap U$, an affine subspace containing $A$ and contained in every such subspace. This proves that the stated affine hull is the smallest one. Its direction space is a subspace of the finite-dimensional $E$, so the finite-dimensional dimension applies. The wall definition selects a nonempty set by its specified intersection and affine dimension, so it makes no assertion that every arrangement hyperplane supports a wall. The gallery condition is a finite condition on already defined chambers and walls. Finally, $L$ is a linear subspace, since intersections preserve zero, addition and scalar multiplication. Hence the quotient construction applies, and the displayed criterion specifies simpliciality using the canonical images $q(v_i)$ in $E\mathrm{/}L$.

For the empty arrangement with $T=E$, the complement is $E$. If $n\ge 1$, ℝⁿ is polygonally connected, connected, locally path-connected and locally connected gives connectedness of ${\mathbb{R}}^n$ with its Euclidean topology. For each $x\in E$ and $r>0$, the bijection $c$ takes $\{y\in E:d_E(x,y)<r\}$ exactly onto $\{v\in {\mathbb{R}}^n:d_2(c(x),v)<r\}$. The ball criterion for openness therefore sends every open subset of $E$ to an open subset of ${\mathbb{R}}^n$. A separation of $E$ into two disjoint nonempty open sets would be carried by $c$ to such a separation of ${\mathbb{R}}^n$, contradicting connectedness (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets). If $n=0$, $E$ is a singleton and cannot be partitioned into two nonempty sets, so it is connected as well. Thus in every dimension $E$ is connected, and it is the sole chamber by the connected-component definition: the union of all connected subsets containing any given point already includes $E$ itself (Connected components, quasicomponents, and totally disconnected spaces). Here $L=E$, $E\mathrm{/}L$ is the zero space with empty basis, and $L+\{0\}=E$, so this chamber satisfies the stated simpliciality criterion.