Tychonoff theorem motivation I am studying for a master's degree and I have to choose a topic and finally I decided to choose. Tychonoff Theorem in General Topology.
Here I studied about Tychonoff Theorem and also I found some easy proof of this theorem. But I am looking for motivation behind this theorem and what is the idea of  this theorem. If I would found some interesting application regarding masters level then it'll be great.
Currently I am following Munkres Topology , Ryszard Engelking and Armstrong basic topology.
Any help will be appreciated.
Thanks in advance!
 A: It would seem that what led Tychonoff to his product theorem was primarily the question of exactly which (Hausdorff) spaces admit nice compactifications.
It is this question which is the focus of his 1930 paper Über die topologische Erweiterung von Räumen where a partial answer was supplied. The main theorem of the paper loosely reads as follows.

For each cardinal $\tau$ there exists a compact Haudorff space $R_\tau$ of weight $\tau$ with the property that any normal space of weight $\leq\tau$ embeds into $R_\tau$. When $\tau=\aleph_0$, the space $R_{\aleph_0}$ is homeomorphic to the Hilbert cube.

He points out that this implies a special case of the Urysohn Metrisation Theorem amongst other things.
The point is of course that the compact space $R_\tau$ is a $\tau$-weighted product of unit intervals, and the hard part of the paper appears in $\S2$, where Tychonoff proves that this product is indeed compact. This would be the very first and rather special case of what would later be called the Tychonoff Product Theorem.
It is amusing that the result here is of such secondary importance to the paper that I cannot even find a lemma or proposition statement to extract to highlight it. Neither is there any mention whatsoever of das Auswahlaxiom, despite the enthusiasm of other commenters here.
Something else which strikes me as odd is that Tychonoff seems to be constructing Stone-Čech type compactifications seven years before either of Stone's or Čech's papers were to be published. Indeed, it is this 1930 paper in which Tychonoff introduced the notion of a completely regular space, and his second main result is that

A space embeds as a subspace of a compact Hausdorff space if and only if it is completely regular.

So, it is all of this which seems to have been the motivation behind Tychonoff's Product Theorem.
The history from here I have not been so successful at tracking down. According to wikipedia Tychonoff pointed out in a 1935 paper of his that the construction given in Über die topologische Erweiterung von Räumen goes through to show that arbitrary products of compact spaces are compact. This would of course be his famous result. I have not been able to find the paper in question, so will decline to comment.
Rest assured, however, that a proof of Tychonoff's Theorem did finally emerge. Actually in the 1937 paper On Bicompact Spaces by Eduard Čech. It is on page 830 in the line starting The Cartesian product.... Here it is used by Čech to construct compactifications of completely regular spaces, and Tychonoff is accredited accordingly.
It would truly seem that the mathematics community did not come to recognise the importance of the Tychonoff Product Theorem until long after it had been accepted into the subject.
A: Hmm, well I have interesting memories of Tychonoff theorem, dating back to when I took Spanier's point-set topology and Intro to functions of a real variable at Berkeley.  He said he knew I didn't get the proof right, because I didn't use the axiom of choice.
But, I can tell you that the Tychonoff theorem, along with Urysohn metrization, is one of the two premiere theorems in general topology.  It says, as you know, that the product of compact spaces is compact.  This distinguishes the product topology from the box topology, since the theorem would not be true in the latter.
Munkres is a good reference.  And there's Kelley.
