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Next: Approximating Compact Hausdorff Spaces Up: Digital Topology Previous: Digital Topology in Computer

Comparing Digital and Metric Spaces

Among the approaches to digital topology discussed in the first section, only Khalimsky's (based on Khalimsky's space $ \mathbb{Z}^n$) is purely topological in that concepts used in digital topology, such as ``connected'' and ``boundary'', are given their topological meanings. It is not difficult to show that the Khalimsky topology on $ \mathbb{Z}$ is (up to homeomorphism) the only one such that the resulting space is a COTS: a connected space such that among any three distinct points is one whose deletion from the space leaves the others in distinct components of the remainder (see [KKM91a]). Intervals in $ \mathbb{R}$ are also COTS.

In this section we explain why any such purely topological approach must be based on this space. Such an explanation, and much more, results from the notion of metric analog, defined in [KK90] by Kong and Khalimsky. Our definition uses the following terminology: As usual, given spaces with base points $ (A,a_0),(M,m_0)$, a (base point preserving) map $ f:(A,a_0)\to(M,m_0)$ is a continuous function such that $ f(a_0)=
m_0$. Given two such maps $ f,g$, a homotopy $ F:A\times[0,1]\to M$ from $ f$ to $ g$ is a continuous function such that, whenever $ a\in A$ and $ t\in[0,1]$, $ F(a,0)=f(a),\ F(a,1)=g(a)$, and $ F(a_0,t)=m_0$.

Given a quotient $ q:M\to X$, and two maps $ f,g:A\to M$ such that $ qf=qg$, a homotopy $ F$ from $ f$ to $ g$ such that $ t\to qF(x,t)$ is constant for each $ x$ is said to ignore $ q$. We say $ f,g$ are $ q$-quotient homotopic (via $ F$) and write $ f\stackrel{q}{\simeq} g$.

Definition 1   A metric analog of a topological space with base point $ (X,x_0)$ is a metric space with base point $ (M,m_0)$, with an open quotient map $ q:M\to X$, such that, if $ (A,a_0)$ is any metric space with base point, then:

Composition by the open quotient $ q$ induces a bijection between the path components of $ M$ and those of $ X$, and this composition induces isomorphisms between the homotopy groups of $ M$ and those of $ X$; that is to say, $ q$ is a weak homotopy equivalence between $ M$ and $ X$.

As a central case of the above, let $ q:\mathbb{R}\to\mathbb{Z}$ be defined by setting, for $ n\in\mathbb{Z},\ q(2n)=2n$ and $ q(x)=2n+1$ whenever $ x\in(2n,2n+2)$; this is clearly an open quotient. Also let $ r_0\in\mathbb{R},z_0\in\mathbb{Z}$, be base points such that $ q(r_0)=z_0$. We show that $ (\mathbb{R},r_0)$, with $ q$, is a metric analog of $ (\mathbb{Z},z_0)$:

To see the two special properties of the definition, note that if $ ((A,d),a_0)$ is a metric space with base point and $ f:A\to\mathbb{Z},$ then the function $ \hat f$, defined by $ \hat f(x)=2n+\frac{2d(x,f\sp{-1}\{2n\})}
{d(x,f\sp{-1}\{2n\})+d(x,f\sp{-1}\{2n+2\})}$ if $ f(x)\in\{2n,2n+1\}$ is continuous; it certainly satisfies $ f=q\hat
f$. For the other property, notice that if $ g$ is another such map and $ qf=qg$, then $ f(x),g(x)$ are always in the same interval of the form $ [2n,2n]$ or $ (2n,2n+2)$, thus $ H(x,t)=(1-t)f(x)+tg(x)$ is also in the same one of these intervals, which is to say that, for each $ x,t$, $ qH(x,t)=qf(x)=qg(x)$. $ H$ is clearly continuous, so a homotopy from $ f$ to $ g$ ignoring $ q$.

The map $ q\sp n:\mathbb{R}\sp n\to\mathbb{Z}\sp n$ also has the above properties, as seen coordinatewise, and similar considerations (see [KKM91b]) show that each Alexandroff $ T_0$ space has a metric analog. The following summarizes key facts about the existence and uniqueness (up to homotopy equivalence) of metric analogs. (Below, let $ \bf 1_A$ denote the identity map on $ A$.)

Theorem 2 (a)   Each $ T_0$ countable join of Alexandroff topologies has a metric analog, so all second countable $ T_0$ spaces have them (see [KKM91b]).

(b) ([KK90]) Suppose $ (M,q)$ is a metric analog of a space $ X$. If $ (N,r)$ is another metric analog of $ X$, then there are maps $ f:M\to N$ and $ g:N\to M$ such that $ gf\stackrel{q}{\simeq}\bf 1_M$ and $ fg\stackrel{q}{\simeq}\bf 1_N$. Conversely, if $ N$ is a metric space, $ r:N\to X$ is an open quotient, and there are maps $ f:M\to N,\ g:N\to M$ so that $ gf\stackrel{q}{\simeq}\bf 1_M$ and $ fg\stackrel{q}{\simeq}\bf 1_N$, then $ (N,r)$ is metric analog of $ X$.

The converse part of (b) is useful in creating other metric analogs from a given one. In particular, it is used to show the existence, for each locally finite space, of a polyhedral analog: one whose space is a geometric realization of the abstract simplicial complex which has a vertex identified with each point in the finite space, and whose simplices are its specialization order chains, with the quotient map which takes each point of this metric space into the specialization-largest vertex of the smallest simplex whose geometric realization contains the point. The proof in [KMW91] of a Jordan surface theorem for three-dimensional digital spaces uses polyhedral analogs.

The definition of 4-adjacency in $ \mathbb{Z}^2$ (6-adjacency in $ \mathbb{Z}^3$) is easily extended to obtain $ 2n$-adjacency in $ \mathbb{Z}^n$, and that of 8-adjacency in $ \mathbb{Z}^2$ (26-adjacency in $ \mathbb{Z}^3$) is easily extended to get $ (3^n-1)$-adjacency in $ \mathbb{Z}^n$. In [Kon], the fact that $ \mathbb{R}^n$ is a metric analog of $ \mathbb{Z}^n$ is used to establish the following result, which is the result we promised at the beginning of the section.

Theorem 3   The Khalimsky topology on $ \mathbb{Z}\sp n$ is, up to translation, the only simply-connected topology on $ \mathbb{Z}^n$ whose connected sets include all $ 2n$-connected sets but no $ (3^n-1)$-disconnected sets.

next up previous
Next: Approximating Compact Hausdorff Spaces Up: Digital Topology Previous: Digital Topology in Computer
benn 2002-04-16