Construct a path-connected metric space and a discontinuous function such that is continuous for any continuous path .

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# Spooky geometry III

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Construct a path-connected metric space and a discontinuous function such that is continuous for any continuous path .

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If is locally path-connected and first-countable, it is true that a function such that its composition with any path is continuous is itself continuous. We will show a counterexample where is locally path-connected but not first-countable.

Let be the subspace of of points of the form for some and natural ; that is, is the union of the unit segments corresponding to the “coordinate axes”.

Let be any linear functional such that and consider its restriction to . Obviously this restriction is discontinuous at 0 since is unbounded. However, the image of any path is contained, by compactness, in a finite-dimensional subspace of the form , so the restriction of to this subspace is bounded, and therefore continuous.

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The above solution is wrong, since the example is both locally path-connected and first-countable (any metric space is first-countable!). In fact, it is not true that the image of a continuous path is contained in a finite-dimensional subspace. For instance, consider a path which goes from to and returns in , then from to and returns in , and so on. This is continuous in by the pasting lemma, and if we set , the fact that implies is continuous at , too. What led to this mistake was the fact that the image of a compact set in a CW-complex intersects at most finitely many cells, but the obvious CW structure on the set does not induce the same topology we are considering: it is not even metrizable, since the complex is not locally finite!

Here is a correct solution, due to Pablo, which is a variation on the comb space. Let and define if and if . This is obviously discontinuous at any point with positive , but is continuous for any path . Notice that this space is

notlocally path-connected.LikeLike

(The usual comb space also works…)

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