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I'm wondering if a very "sparse" set in a normed vector space can look connected in the weak* topology. Specifically,

Let V be a Banach space, V* its dual, and X a (uncountable) subset of the unit ball in V* such that any two points in X are distance at least 1/2 apart from each other. Now forget the norm on V* for a moment and endow it with the weak* topology instead. Could X be connected? Could it be weak*-dense in the unit ball? Could it contain, say, the image of a closed interval under a non-constant, continuous (with respect to the weak* topology on V*, of course) map?

Interesting examples of what can or can't happen are welcome. FWIW, the space that I was looking at that prompted this question comes from bounded cohomology of groups, but finding interesting examples in, say, $L^\infty[0,1]$ would be fine by me.

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    $\begingroup$ Take $V=C[0,1]$. $[0,1]$ lives in $(V^*,w^*)$ as pointwise evaluations but is discrete in $(V^*,\|\cdot \|)$. $\endgroup$ – Bill Johnson Feb 13 '14 at 16:41
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(This post seems to have been deleted automatically, although a functional analyst has told me that the question seems reasonable. I am therefore undeleting and promoting Bill Johnson's comment to an answer. He is cordially invited to answer himself, whereupon this answer can be deleted.)

"Take $V=C[0,1]$. $[0,1]$ lives in $(V^*,w^*)$ as pointwise evaluations but is discrete in $(V^*,\|\cdot \|)$."

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