{"id":1677,"date":"2019-02-20T14:56:44","date_gmt":"2019-02-20T13:56:44","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1677"},"modified":"2022-11-19T15:08:38","modified_gmt":"2022-11-19T14:08:38","slug":"a-core-compact-non-locally-compact-space","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1677","title":{"rendered":"A core-compact, non-locally compact space"},"content":{"rendered":"

Last time<\/a>, I announced that we would do\u00a0Exercise V-5.25 of the red book [1], following Hofmann and Lawson [2], in order to show that there exists a space\u00a0X<\/em> that is core-compact but not locally compact. \u00a0This is pretty lengthy, but here we go.<\/p>\n

We first build a nice space\u00a0Y<\/em>, out of which we will carve out the stranger space\u00a0X<\/em>.<\/p>\n

The space Y<\/em><\/h2>\n

Let I<\/strong> be the closed interval [0,1], with its usual\u00a0metric topology. \u00a0Let J<\/strong> be\u00a0the half-open interval [0,1) with the Scott topology of the reverse ordering \u2265: its open subsets are [0, a<\/em>) with 0\u2264a<\/em>\u22641. \u00a0Then, we let Y<\/em> be I<\/strong>\u00a0\u00d7 J<\/strong>, with the product topology.<\/p>\n

This is really the space\u00a0Y<\/em> that is used in\u00a0Exercise V-5.25 of [1], although you may fail to recognize it. \u00a0I will make one step in that direction when I talk about lower semicontinuous maps below. \u00a0You can infer the rest if you do Exercise V-5.23 of [1].<\/p>\n

For now, we recognize that\u00a0Y<\/em> is sober, being a product of sober spaces (Theorem 8.4.8 in the book<\/a>). \u00a0Indeed,\u00a0I<\/strong> is Hausdorff, and\u00a0J<\/strong> is a continuous dcpo (where\u00a0a<\/em> is way-below\u00a0b<\/em> if and only if\u00a0a<\/em>>b<\/em>), so both are sober by Proposition 8.2.12 of the book<\/a>.<\/p>\n

Y<\/em> is locally compact, too, because it is a finite product of locally compact spaces (Proposition 4.8.10). \u00a0I<\/strong> is locally compact because it is compact Hausdorff (see Example 4.4.3 and use Proposition 4.8.8), and\u00a0J<\/strong> is locally compact because every continuous dcpo is in its Scott topology (Corollary 5.1.36).<\/p>\n

Lower semicontinuous maps<\/h2>\n

It will be useful to observe that, given any open subset\u00a0W<\/em> of\u00a0Y<\/em>, we can define a map\u00a0f<\/em> :\u00a0I<\/strong>\u00a0\u2192\u00a0[0,1] by\u00a0f<\/em>(x<\/em>)=sup {y<\/em> | (x<\/em>,y<\/em>)\u00a0\u2208\u00a0W<\/em>}, and that this map is lower semicontinuous (i.e., continuous from\u00a0I<\/strong> to [0,1] with the Scott topology of the usual ordering \u2264).<\/p>\n

Indeed,\u00a0f<\/em>-1<\/sup>((a<\/em>, 1])={x<\/em> |\u00a0\u2203y<\/em>>a<\/em>,\u00a0(x<\/em>,y<\/em>)\u00a0\u2208\u00a0W<\/em>}. \u00a0Writing\u00a0W<\/em> as a union of open rectangles Ui<\/sub><\/em>\u00a0\u00d7<\/strong> Vi<\/sub><\/em>, we see that\u00a0f<\/em>-1<\/sup>((a<\/em>, 1]) is equal to the union of all Ui<\/sub><\/em>\u00a0such that Vi<\/sub><\/em>\u00a0contains some\u00a0y<\/em>>a<\/em>, hence is open in\u00a0I<\/strong>.<\/p>\n

Conversely, given any lower semicontinuous map\u00a0f<\/em> from\u00a0I<\/strong> to [0,1], we can define an open set\u00a0W<\/em> as {(x<\/em>,y<\/em>) \u2208 I<\/strong>\u00a0\u00d7 J<\/strong>\u00a0| y<\/em><f<\/em>(x<\/em>)}. \u00a0This is open because it is the union over all\u00a0a<\/em> of f<\/em>-1<\/sup>((a<\/em>, 1])\u00a0\u00d7<\/strong>\u00a0[0,\u00a0a<\/em>).<\/p>\n

Finally, the two constructions are inverse of each other: starting from an open set\u00a0W<\/em>, building\u00a0f<\/em> and going back, we obtain\u00a0{(x<\/em>,y<\/em>) \u2208 I<\/strong>\u00a0\u00d7 J<\/strong>\u00a0| y<\/em><sup {y’<\/em> | (x<\/em>,y’<\/em>)\u00a0\u2208\u00a0W<\/em>}}, which is just\u00a0W<\/em>, because\u00a0(x<\/em>,y<\/em>) \u2208\u00a0W<\/em> if and only if there is a\u00a0y’<\/em>>y<\/em> such that (x<\/em>,y’<\/em>) \u2208\u00a0W<\/em>; and starting from a lower semicontinuous map\u00a0f<\/em>, building\u00a0W<\/em> and going back, we obtain a function that maps every\u00a0x<\/em> to\u00a0sup {y<\/em> | (x<\/em>,y<\/em>)\u00a0\u2208 {(x<\/em>,y<\/em>) \u2208 I<\/strong>\u00a0\u00d7 J<\/strong>\u00a0| y<\/em><f<\/em>(x<\/em>)}}=f<\/em>(x<\/em>).<\/p>\n

This defines an order isomorphism between\u00a0O<\/strong>(Y<\/em>) and LSC(I<\/strong>,[0,1]), the set of lower semicontinuous maps from\u00a0I<\/strong> to [0,1], with the pointwise ordering. \u00a0This will be useful in the next section.<\/p>\n

The space X<\/em><\/h2>\n

We pick a Bernstein subset<\/a> A<\/em> of\u00a0R<\/strong>. \u00a0We now define\u00a0X<\/em> as the subspace of\u00a0Y<\/em> of points (x<\/em>,y<\/em>) such that:<\/p>\n