# Like a cell complex but characteristic maps send n-cell boundary to other n-cells

Recall that a cell complex is a topological space $$X$$ that satisfies the following definition:

$$X$$ decomposes as a union of open cells of varying dimensions. For each cell $$C$$ of dimension $$n\geq 1$$ there is a continuous map $$f\colon D^n \to X$$ (called the characteristic map of $$C$$) that restricts to a homeomorphism $$(D^n)^{int} \to C$$ and maps $$\partial D$$ into a union of cells in $$X$$ of dimension $$.

I have a topological space that by construction satisfies all the parts of this definition except for one: it has a decomposition into cells and each cell has an obvious characteristic map, but the characteristic maps map the boundary of each n-cell into a union of other n-cells (in fact, all the cells in the space I am considering are of the same dimension $$\geq 1$$).

Does this variation of cell complex have a name? Is there a straightforward way to turn this space into a cell complex?

Edit: I changed the question from being about CW complexes to being about cell complexes. The space in question does satisfy axioms (C) and (W) from Lee, given here, which is the reason I originally asked about CW complexes: the space satisfies the definition of a CW complex (following Lee) except for one part of the definition of a cell complex.

• The question is not really precise. You can build a space in finitely many steps by attaching cells in the way you decribe - but what does this mean? Start with one $0$-cell and attach other $0$-cells as described? Then the $0$-skeleton always is a singleton. Or do you allow to start with an arbitrary discrete space? Similarly in higher dimensions: You must attach at least one $n$-cell to the $(n-1)$-skeleton. – Paul Frost Oct 27 '18 at 23:25
• @PaulFrost I realize, in trying to answer your question, that my question is really about the definition of cell complexes, not CW complexes. Now it should be perfectly clear. – AnonymousCoward Oct 28 '18 at 10:26
• You write "the characteristic maps map the boundary of each $n$-cell into a union of interiors of other $n$-cells". What do you mean by the interior of an $n$-cell? Just a normal open $n$-cell or the topological interior of a closed $n$-cell? – Paul Frost Oct 28 '18 at 10:38
• @PaulFrost Of course, the cells are open, so the interior of a cell is the cell itself. I'm sorry for adding the unnecessary extra word. I removed it. – AnonymousCoward Oct 28 '18 at 10:42
• Can you give a concrete example (say, with two cells)? – Paul Frost Oct 28 '18 at 14:06