Diskussion:Abstrakter Zellkomplex

Hausdorff-Raum

Es trifft nicht zu, dass in einem Hausdorff-Raum die Umgebung eines Punktes unendlich viele Punkte enthalten muss. Gegenbeispiel ist z.B. eine endliche Menge mit der diskreten Topologie. --Suhagja (Diskussion) 05:55, 16. Apr. 2013 (CEST)

Materialsammlung

Steinitz

Bei Steinitz findet sich folgende Definition des Begriffs "Polyedrische Mannigfaltigkeit"":

1. Eine polyedrische Mannigfaltigkeit ist ein endliches System von Elementen.

2. Jedem Element a wird eine Dimension [a] zugeschrieben; dieselbe ist gleich einer der Zahlen 0,1,...,n.

3. Zwei Elemente a,b heissen entweder inzident (in Zeichen (a,b)=1 oder (b,a)=1) oder nicht inzident ((a,b)=0 oder (b,a)=0).

4. Ist [a]=[b], so ist dann und nur dann (a,b)=1, wenn a=b ist; ist (a,b)=1,(b,c)=1, [a]≥[b] ≥[c], so ist (a,c)=1.

5. Jedes Element erster Dimension (Strecke) ist mit zwei Elementen 0ter Dimension (Ecken), jedes Element (n-1)ter Dimension mit ein oder zwei Elementen nter Dimension inzident.

und dann noch 5 weitere Axiome.

Tucker

An abstract cell complex is a collection, K, of objects, xi, called abstract cells. To each cell xi is assigned an integer p called its dimension. and we write xi_p if the dimension is to be explicit. An order relation < is given in K (we say xi is a face of xi' if xi<xi') and to each pair of cells xi,xi' of consecutive dimension there is defined an integer [xi,xi'] called the incidence number.

AC1: < is a strict partial ordering.

AC2: xi_p<xi'_q and xi_p\not=xi'_q imply p<q.

AC3: For each pair, xi, xi there exist only a finite number of cells xi' with xi<xi'<xi.

AC4: [xi_p,xi'_{p+1}]\not=0 implies xi_p<xi'_{p+1}.

AC5: For each pair xi_{p-1},xi_{p+1} \sum_{xi'_p}[xi_{p-1},xi'_p][xi'_p,xi_{p+1}]=0.

Kovalevsky

An abstract cell complex (ACC) C = (E,B, dim) is a set E of abstract elements provided with an antisymmetric, irreflexive, and transitive binary relation B ⊂ E × E called the bounding relation, and with a dimension function dim : E −→ I from E into the set I of non-negative integers such that dim(e�) < dim(e��) for all pairs (e�, e��) ∈ B.

The bounding relation B is a partial order in E. The bounding relation is denoted by e� < e�� which means that the cell e� bounds the cell e��. Furthermore the property that any cell can only bound cells of higher dimension is emphasized by this notation: e� < e�� =⇒ dim(e�) < dim(e��)

If a cell e� bounds another cell e�� then e� is called a side of e��. The sides of an abstract cell e�� are not parts of e��. The intersection of two distinct abstract cells, different from that of Euclidean cells, is always empty.

If the dimension dim(e�) of a cell e� is equal to d then e� is called d-dimensional cell or a d-cell. An ACC is called k-dimensional or a k-complex if the dimensions of all its cells are less or equal to k. Cells of dimension k, which means cells bounding no other cells, are called base cells.

In the field of digital image processing we normally use regular imagecarrier. There the 0-cells are called points, 1-cells are called cracks, 2-cell are called pixels and 3-cells are called voxels.

The central facts of definition 5 can be summarized in three axioms called the cell complex axioms [14]:

(C1) From (e�, e��) ∈ B and (e��, e���) ∈ B follows (e�, e���) ∈ B (transitivity)

(C2) From (e�, e��) ∈ B follows dim(e�) < dim(e��) (monotony)

(C3) For each element e� there exist only a finite number of elements e�� with (e��, e�) ∈ B

Historisches

Vielleicht eine brauchbare Quelle zur Geschichte des Begriffs: https://researchspace.auckland.ac.nz/bitstream/handle/2292/2723/CITR-TR-60.pdf?sequence=1

Zellenkomplex

Gibt es irgendein Quellenbeispiel fur die Bezeichnung "Zellenkomplex" statt "Zellkomplex"? Sonst losche ich diese Bemerkung.--Suhagja (Diskussion) 11:10, 20. Apr. 2013 (CEST)