If \(Y\) is any subset of a topological space \(X\), then \(\dim Y \leq \dim X\).
If \(X\) is covered by open subsets \(\ts{U_i}\), then \(\dim X = \sup_i \dim U_i\).
Give an example of a topological space \(X\) with a dense open subset \(U\) such that \(\dim U < \dim X\).
If \(Y\) is a closed subset of an irreducible finite-dimensional space \(X\) with \(\dim Y = \dim X\), then \(Y = X\).
Give an example of a Noetherian topological space of infinite dimension.
Solution
(a) Let \((U_i)_{0 \leq i \leq n}\) be a strict chain of irreducible closed subsets of \(Y\), so \(U_i = V_i \intersect Y\) for some \(V_i\) closed in \(X\). The \(V_i\) need not form a chain; take partial unions \(\tilde{V}_n = \Union_{i \leq n} V_i\), which do, and which remain strict. So \(\dim X \geq n\).
(b) By (a) it suffices to find one \(U_i\) with \(\dim U_i \geq \dim X\). Take a strict chain \((C_j)_{0 \leq j \leq n}\) witnessing \(\dim X = n\) and choose \(i_0\) with \(U_{i_0} \intersect C_0 \neq \emptyset\). Then \((C_j \intersect U_{i_0})_j\) is a strict chain of length \(n\) in \(U_{i_0}\).
(c) \(X = \AA^1_\CC = \Spec \CC[x]\) has dimension \(1\); the open subset given by the generic point alone has dimension \(0\) and is dense.
(d) Take a chain \((C_i)_{0 \leq i \leq n}\) in \(Y\) with \(n = \dim Y = \dim X\), arranged so that \(C_n = Y\). If \(Y \subsetneq X\) then \(C_0 \subsetneq \cdots \subsetneq C_n = Y \subsetneq X\) is a strict chain of length \(n+1\) in \(X\), since \(X\) is irreducible and therefore itself an irreducible closed subset. That contradicts \(\dim X = n\).
(e) Take \(X = \ZZ_{\geq 0}\) with closed sets \(\emptyset\), \(X\), and \(\ts{1,\ldots,i}\) for each \(i\). The chain \(\ts{1} \subsetneq \ts{1,2} \subsetneq \cdots\) is infinite, so \(\dim X = \infty\), and every descending chain of closed sets is finite, so \(X\) is Noetherian.
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