Order dimension and topological dimension

Hello,

Let X be a set and R be an ordering relation on X. Then C(X,R) denotes
the set of linear ordering relations containing R. It is well-known
that C(X,R) is non-empty (Szpilrajn's theorem) and that the
intersection of C(X,R) equals R.

The order dimension of (X, R) is defined to be

min { card(K) : K \subseteq C(X,R) and \bigcap K = R }.

Is there a connection / relationship between the order dimension of a
poset and the topological covering dimension of X endowed with the
interval topology?

Thanks, Dominic

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