Re: 'local' metric space

On 07-10-2007 16:38, sanchopancho80@xxxxxx wrote:

let X be a topological space, such that every point has got an open
neighbourhood, which is a metric space (perhaps one might call this a
'local metric space', the topology on X is not necessarily induced by
one of the metrics).
Is it true that if X is connected, then X is a metric space in
general?
No. The long line is a connected local metric space, but is not
metrizable.

oh yes, that's right. Does anything like this appear in the
literature? I haven't found anything about 'local metric spaces'.

But you will find a lot about "locally metrizable spaces". See, among
many other examples, M. R. Koushesh's article "On one-point metrizable
extensions of locally compact metrizable spaces", which was published
this year at Topology and Applications (vol. 154, no. 3, 698-721).

Best regards,

Jose Carlos Santos
.

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