Friday, February 19, 2016

Exponential Law in Topology


Exponential Law in Topology


Exponential law is well known in the context of classical algebra, namely $(a^b)^c = a^{bc}$.
In  the set-theoretic context maps between from a set $X$ to a set $Y$ are often denoted by $Y^X$. Is this a coincident? It turns out  that not only that  cardinality of corresponding sets can be expressed by power relation $\left|Y^X \right| = |X|^{|Y|}$ , but it also has a certain form of exponential law: $(Z^X)^Y \cong  Z^{X \times Y} $  or in type theoretic $a$ notation (which I personally find way more convenient) $Y \to X \to Z \cong  X * Y \to Z$.

 To prove our last statement it is sufficient to mark out natural bijection $I$:$$ \begin{array} I I & :: & (Y \to X \to Z) \to (X * Y\to Z) \\ I(f)(x,y) &=& f(y)(x) \\I^{-1}(g)(y)(x) &=& g(x,y) \\ \end{array} $$ However, it must be noted that that this fact is established for now only for $\mathsf{SET}$ Category (have we mentioned any others?). As an uncultured tramp I was stroked by the fact that in the category $\mathsf{TOP}$  if we will view $X \to Z$ as space of continuous functions endowed with compact-open topology exponential law holds only then  $X$ is locally compact and Hausdorff (or just compact if you are an algebraic geometrician) and $Y$ just Hausdorff (this conditions turns out to be sufficient. but not necessary) . At first it seemed unbelievable that natural bijection is able to distort continuity properties of maps.   Actually this made my barbaric uncultured blood so hot that that I spent several pointless hours trying to disprove this result.

As it turned out letter, this problem also bothered many topologists in the past.  For example, Edwin Spanier proposed concept of quasitopology  for overcoming this issue.  


Edwin Spainer (1921 -1996)

 Problem of exponential law in topological categories found its development in many other works also. The modern state of the subject can be checked out at this nLab article.

The morale of the story is that this was a test blog entry. And possibly there will be more.
     


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