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 Post subject: On The Dimension Of A Topological SpacePosted: Wed Apr 13, 2016 9:28 am
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Joined: Tue Nov 10, 2015 8:25 pm
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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 $Y = X$.

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