Three things I don't get about CW-complexes

[quasar987]

(1) We say a CW-complex is finite if its cell decompositions are finite. Why is this well defined? I.e., why can't a CW-complex admits a decomposition into a finite number of cells and another decomposition into an infinite number of cells?

(2) We say a CW-complex X has dimension n if n is the dimension of the cell of X of highest dimension. Why is this well defined? I.e., why can't a CW-complex admits a cell decomposition where the highest dimensional cell has dimension n and another cell decomposition where the highest dimensional cell has dimension m? I'm guessing this can be shown using Brouwer's invariance of domain, but I've been having technical problems implementing this idea..

(3) Given a CW-complex X, a subcomplex of X can be defined as a subspace A of X which is the reunion of closed cells of X. Why is a subcomplex necessarily closed in X?

Thanks for any help.

 

[wofsy]

(1) We say a CW-complex is finite if its cell decompositions are finite. Why is this well defined? I.e., why can't a CW-complex admits a decomposition into a finite number of cells and another decomposition into an infinite number of cells?

(2) We say a CW-complex X has dimension n if n is the dimension of the cell of X of highest dimension. Why is this well defined? I.e., why can't a CW-complex admits a cell decomposition where the highest dimensional cell has dimension n and another cell decomposition where the highest dimensional cell has dimension m? I'm guessing this can be shown using Brouwer's invariance of domain, but I've been having technical problems implementing this idea..

(3) Given a CW-complex X, a subcomplex of X can be defined as a subspace A of X which is the reunion of closed cells of X. Why is a subcomplex necessarily closed in X?

Thanks for any help.

1) A finite CW complex is compact An infinite one is not.

2) In the top dimension the CW complex contains open sets homeomorphic to R_n not R_n+1.

3) by definition the closure of each open cell in a sub-complex is contained in the sub-complex