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(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.