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On The Dimension Of A Topological Space

Posted: Wed Apr 13, 2016 9:28 am
by Tsakanikas Nickos
Let $X$ be a topological space. Show that
  • If $Y$ is any subset of $X$, then $\dim Y \leq \dim X$.
  • If $ \{ U_{i} \}_{i \in I} $ is an open covering of $X$, then $\dim X = \sup \dim U_{i}$
Now, suppose that $X$ is irreducible and finite-dimensional. Show that if $Y$ is a closed subset of $X$ such that $\dim Y = \dim X$, then $Y = X$.