In Hartshorne's book definiton of a dimension is given as follows:(adsbygoogle = window.adsbygoogle || []).push({});

İf X is a t.s. , dim(X) is the supremum of the integers n s.t. there exist a chain

[tex] Z_0 \subsetneq Z_1.....\subsetneq Z_n[/tex]

of distinct irreducible closed subsets of X

My question is:

Can we conclude directly that any topological space has dim greater than or equal to 1, since empty set and and X itself is always closed?

Example in the same book says no in a way.It says A^1 has dim 1.

Although [tex] \emptyset \subsetneq {any point} \subsetneq X [/tex]

Should I exclude empty set ?

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# Dimension of a topological space

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