2,407 reputation
1028
bio website
location sqrt(-1)
age
visits member for 4 years, 2 months
seen 3 hours ago

 
  two queens
 
 
    mathematics and death
    not cheerful not sad
    guide me through the land
    pat me on my head
 
    they live
    in my one man nation
    i follow them
    to my destination
 
 
wh,
(long ago)



  polynomials over finite Galois field
  spread so evenly across their finite affine space
  i wish for a network of friends
  to count on


wh,
1996-03-05/06



20h
comment How many mathematicians are there?
There are 831 MO user pages + 5 participants on page 832. Thus the number of mathematicians is: $\ 831*36 + 5\ =\ \mathbf {29921}.\ \ $ That's it! :-)
20h
comment How many mathematicians are there?
There are mathematicians in different sense: 1. mth. PhD 2.mth research publication 3.referred to by MR 4.listed by MGP ... There are errors, they partially cancel out, one gets an idea in one way or another. I'd also consider a recognized mathematician: independently refereed to her/his mth research by 10 or more different mathematicians (this is not circular).
1d
comment Topological characterization of injective metric spaces
Monomorphism of the metric category of metric spaces are not isometric embeddings (they are just injective metric maps). Thus a pure categorist should take this into account.
1d
comment Topological characterization of injective metric spaces
Isbell introduced (1) injective metric envelope, and (2) proved that the metric envelope of a Banach space is virtually the same as the Banach injective envelope (I rediscovered both a little later).
1d
comment Topological characterization of injective metric spaces
@Ali, there are no non-trivial projective objects in the metric category of all metric spaces (only the empty space and singletons are projective). There are no non-trivial projective objects in the metric category of all bounded metric spaces. And in the metric category of all spaces of diameter $\le 1,\ $ a space $\ (X\ d)\ $ is projective $\ \Leftarrow:\Rightarrow\ d$ is the $\ 0\!-\!1$ metric.
1d
revised Topological characterization of injective metric spaces
non-mathematical detail
1d
comment Topological characterization of injective metric spaces
Thank you @Tom. Isbell's paper is a well known classic. Of course the equivalence of the ball binary property, metric injectivity and metric absolute retract property is a very simple property (even if it can be formulated as a theorem, why not). Of course being a an absolute (topological) retract among the metrisable spaces is more general than being homeomorphic to an injective metric space (even among the separable metric spaces, or even among finite polyhedra) these two notions are not equivalent.
1d
revised Topological characterization of injective metric spaces
typo
1d
revised Topological characterization of injective metric spaces
A clearer(?) question.
1d
revised Topological characterization of injective metric spaces
A clearer(?) question.
1d
revised Topological characterization of injective metric spaces
typos
1d
comment Topological characterization of injective metric spaces
Thank you, @Andrej, for the reference. It is not related (directly) to the question of characterization, it is not even mentioned as long as I know (actually, I've never saw it mentioned). This E&K's text belongs to the school represented for a long time by W.A.Kirk, which is concerned with non-expanding (i.e. metric) maps of metric spaces and their fixed point property--one could say, with the difficult case of this kind of the fixed point property as opposed by the Banach fpp theorem which can be considered the easy case.
1d
comment Topological characterization of injective metric spaces
Equivalence of A&P's binary intersection property and metric injectivity is about the earliest and most fundamental; I've written in my OM-QUestion from the start: Injective metric spaces were introduced ... under the hyper-convex spaces, via a binary intersection property of closed balls. Equivalently, a metric space (Z ρ) is called injective .... Thus this much should be clear from the QUESTION statement. (will continue below)
1d
comment Special retraction from a metric space onto an arc
This is Hausdorff theorem about extending an equivalent metrics from a closed subset onto the whole space (Toruńczyk--much later of course--obtained a simple and elegant proof). This was mentioned in another thread on MO.
1d
revised Topological characterization of injective metric spaces
metric spaces tag
1d
revised Topological characterization of injective metric spaces
necessary details missing (the same sense).
1d
comment Topological characterization of injective metric spaces
Just in case, don't worry about spaces which are not complete--they are never injective. A non-complete space is not a retract of its completion.
1d
asked Topological characterization of injective metric spaces
1d
revised Special retraction from a metric space onto an arc
An additional remark
1d
answered Special retraction from a metric space onto an arc