I'm trying to prove the decomposition theorem for the topological dimension:

This is a known result (eg Engelking) only for a non-empty separable metric space $ X $, the small inductive dimension $ indX $ satisfies the decomposition theorem:

$ indX leq n $ if and only if $ X = Y cup Z $ with $ indY leq n-1 $ and $ indZ leq0 $

We also know that in the case of separable metric spaces, the small inductive dimension and the topological dimension (also called the Lebesgue recovery dimension) are equivalent. I am given the following definition for the topological dimension:

Let $ alpha = (A_i) _ {i in I} $ to be an open blanket of $ X $ for some indexes $ I $. Given $ x in X $, to define $ ord_x ( alpha) = – 1 + # {{ in I: x in A_i $. Now define $ ord ( alpha) = sup_ {x in X} ord_x ( alpha) $. We say that $ beta = (B_j) _ {j in J} $ is a finished open top finished finer than $ alpha $ if for all $ j in J $ it exists $ i in I $ with $ B_j subset A_i $. To define $$ D ( alpha): = min _ { beta} ord ( beta) $$ or $ beta $ is finer than $ alpha $. Finally, the topological dimension of $ X $ is given by $$ dim_tX: = sup_ alpha D ( alpha), $$ or $ alpha $ going on all the finished open covers of $ X $.

I am now trying to prove the above theorem using the definition of the topological dimension, that is to say that $ dim_tX leq n $ if and only if $ X = Y cup Z $ with $ dim_tY leq n-1 $ and $ dim_tZ leq $ 0. The right-to-left direction is not hard to prove. However, the direction where we are given $ dim_tX leq n $ and must build $ Y $ and $ Z $ causes me problems. I have tried the following:

Let $ alpha $ to be an open blanket over for $ X $ such as $ D ( alpha) = dim_tX = n $ and $ beta $ an open cover finished finer than $ alpha $ with $ n = ord ( beta) $. Consider $ I: {x in X: ord_x ( beta) = n } $. Then I tried to do something with $ X setminus I $ who, hopefully $ dim_t (X setminus I) leq n-1 $ and I tried to pretend that $ I $ will be totally disconnected (or at least that I can find finer open covers of finer and finer $ beta_k $ such as $ I _ { beta_k}: = {x in X: ord_x ( beta_k) = n } $ has the property that $ I _ { beta_k} subset I _ { beta_ {k-1}} $ so what $ I: = bigcap_ {k in mathbb {N}} I _ { beta_k} $ totally disconnected), which is not very clear (and I do not know how to prove, if he is not already wrong to start). I've also thought that it might be helpful to use closed blankets $ beta $ instead of open ones (since the definition of the topological dimension gives the same result), but again, I could not go anywhere.

In total, I'm stuck and I do not know how to prove the direction.