Geslotencell

Geslotencell, or closed cell, is a foundational concept in topology, especially in the study of cell complexes such as CW complexes. In its simplest terms, a closed n-cell is a space that is homeomorphic to the closed n-dimensional unit ball D^n in R^n. The interior of this disk, the open n-ball B^n, corresponds to an open n-cell, while the entire disk maps onto a closed n-cell e in a larger space X. The image of the boundary S^{n-1} forms the cell’s boundary within X.

In a CW complex, each n-cell e has an attaching map φ: D^n → X. The image φ(D^n) is a closed n-cell in X, and the interior φ(int D^n) is the open n-cell that sits inside X. The boundary φ(S^{n-1}) lies in the union of lower-dimensional cells, and the closure of an open cell is the corresponding closed cell. Closed cells are compact, since D^n is compact, and their intersections are governed by the CW structure: different open cells meet only along their closures, which are built from lower-dimensional cells.

Examples include the standard CW structure on spheres, where S^n consists of one 0-cell and one n-cell, with the closed n-cell being D^n and its boundary S^{n-1} attached to the 0-skeleton. Another example is a torus formed by attaching a 2-cell to a 1-skeleton along its boundary.

Geslotencell is often discussed in relation to attaching maps, characteristic maps, and cellular homology, where closed cells provide the basic building blocks for decomposing spaces into comprehensible pieces.