Cells are pasted together, without overlap, to build a space S. For instance, the closed ball in three dimensions consists of a 3 cell (its interior), a 2 cell (its boundary sans the north pole), and a 0 cell (the north pole). Notice that the 2 cell is given the subspace topology first, as defined by S, then it becomes equivalent to an open disk in the plane. Of course there are rules on how these cells are put together.
A point is its own closure, so there are no requirements on f when the cell is a single point.
f maps a closed line segment, the next ball up, onto something that looks like an open interval, and two points, or one common end point (a loop).
Verify that Sn is itself a cw complex. The two cells are the interior of Dn, wrapped around Sn without the north pole, and the north pole. The characteristic function f wraps the disk around the sphere and maps the boundary onto the north pole. This satisfies the requirements of f, and establishes the cw complex. Note that f need not be one to one on the boundary, though it is continuous.
Verify that Dn is also a cw complex. The three cells are the interior, the punctured surface, and the north pole.
A simplex is also a cw complex, since it is homeomorphic to a closed ball; but we usually select a different decomposition. A line segment is a one dimensional simplex, and it consists of a 1 cell (the open interval) and two 0 cells (the end points). Moving to the 2 simplex, i.e. the triangle, let a 2 cell consume the interior, while 1 cells and 0 cells build the boundary. The characteristic function f maps the 2 disk onto the triangle homeomorphically, with the interior mapping to the interior. (In this case f is homeomorphic on all of Dn, including its boundary.) Proceed by induction on the dimension of the simplex, and every simplex is a cw complex.
Paste simplexes together, and every simplicial complex is a cw complex, with a canonical decomposition. Each simplex at each dimension, solid, face, edge, vertex, etc, implies a cell, i.e. the interior of that simplex. Cells and simplexes correspond. The characteristic function maps the boundary of the ball to the boundary of the simplex, which is always finitely many simplexes, or cells, at lower dimensions.
If T is a closed subspace of S, intersect T with any closed cell and find a closed set. The converse holds when the cw complex is finite. Suppose T is not closed in S. Consider a point b in S whose neighborhoods always intersect T. In other words, b is a limit point of T. Intersect T with each open cell in turn, and b is a limit point of one of these intersections, else the union of these (finitely many) intersections leaves b isolated from T. Call this cell E, so that b is a limit point of T∩E. Thus b is a limit point of E, and is contained in the closure of E. Now the intersection of E closure with T does not contain b, and is not closed in S. This is a contradiction, hence T is closed after all.
Let's see what goes wrong when the complex is infinite. Let S be the nonnegative real line. The 0 cells are the integers and the reciprocal integers, and the 1 cells are the open intervals delimited by adjacent points. The positive real line is open in S, but its intersection with each closed cell is closed. We can tweak the topology to fix this problem. Let 0 be an open set all by itself, and S becomes closed.
By definition, a cw complex must satisfy the closed subspace criterion described above. Thus our example does not become a cw complex until we isolate 0.
If S is a simplicial complex, the closed subspace criterion is never an issue, because a set is closed in S iff it is closed in each simplex of S. This falls out of the definition of a simplicial complex as a quotient space of the disjoint union of the individual simplexes.