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I am reading through Stephen Woolard's General topology and I am not sure what he means by "compatible".

The context is as follows. $X$ is a topological space space, and $\delta$ is a proximity on $X$. I am confused by what is meant when he says:

$\delta$ is compatible with the topology on $X$.

Does he mean that for any two subsets $A,B\subseteq X$ we have $A\delta B \iff \overline{A}\cap \overline{B} \neq \emptyset$? (here $\overline{A}$ is the closure of $A$ in the topology).

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    $\begingroup$ Do you mean Stephen Willard's General Topology or is there an actually topology book by Stephen Woolard? $\endgroup$
    – bof
    Sep 23, 2017 at 20:34
  • $\begingroup$ I dk. But the 8th (and last) chapter of General Topology by R. Engelking is entirely about Proximity Spaces. $\endgroup$
    – DanielWainfleet
    Sep 23, 2017 at 21:37

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He means the following: a proximity $\delta$ induces a closure operator: $\overline{A} = \{x: \{x\}\delta A\}$ and this defines a topology in the usual way (closed sets are sets equal to their closure, open sets are their complement etc. ) If this topology equals the original topology on $X$, this topology is compatible with the proximity.

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  • $\begingroup$ I was thinking it was this! Thanks. $\endgroup$
    – fosho
    Sep 23, 2017 at 21:48

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