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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,
19960305/06
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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! :) 
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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). 
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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. 
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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). 
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Topological characterization of injective metric spaces
@Ali, there are no nontrivial projective objects in the metric category of all metric spaces (only the empty space and singletons are projective). There are no nontrivial 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. 
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Topological characterization of injective metric spaces
nonmathematical detail 
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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. 
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Topological characterization of injective metric spaces
typo 
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Topological characterization of injective metric spaces
A clearer(?) question. 
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Topological characterization of injective metric spaces
A clearer(?) question. 
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Topological characterization of injective metric spaces
typos 
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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 nonexpanding (i.e. metric) maps of metric spaces and their fixed point propertyone 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. 
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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 OMQUestion from the start: Injective metric spaces were introduced ... under the hyperconvex 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) 
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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 (Toruńczykmuch later of courseobtained a simple and elegant proof). This was mentioned in another thread on MO. 
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Topological characterization of injective metric spaces
metric spaces tag 
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Topological characterization of injective metric spaces
necessary details missing (the same sense). 
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Topological characterization of injective metric spaces
Just in case, don't worry about spaces which are not completethey are never injective. A noncomplete space is not a retract of its completion. 
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asked  Topological characterization of injective metric spaces 
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Special retraction from a metric space onto an arc
An additional remark 
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answered  Special retraction from a metric space onto an arc 