Hawking in A Brief History of Time: No beginning or end of the Universe I am reading Stephen Hawking's book - A Brief History of Time. I haven't studied cosmology and the related sciences. Nevertheless, I am interested to know few things regarding the extract taken from his book:

On the observational side, by far the most important development has
  been the measurement of fluctuations in the cosmic microwave
  background radiation by COBE (the Cosmic Background Explorer
  satellite) and other collaborations. These fluctuations are the
  finger-prints of creation, tiny initial irregularities in the
  otherwise smooth and uniform early universe that later grew into
  galaxies, stars, and all the structures we see around us. Their form
  agrees with the predictions of the proposal that the universe has no
  boundaries or edges in the imaginary time direction; but further
  observations will be necessary to distinguish this proposal from other
  possible explanations for the fluctuations in the background. However,
  within a few years we should know whether we can believe that we live
  in a universe that is completely self-contained and without beginning
  or end. 



*

*One of the proposals mentioned is that the universe has no boundaries or edges in the
imaginary time direction. How far as of now is this true? Do we live in open space with no boundaries?

*Secondly, I want to know whether the universe in which I am living is enclosed or not?

*What does the author mean when he says that 

we live in a universe that is completely self-contained and without beginning or
  end.

Please clarify by putting forward some analogies.
 A: Hawking was a great physicist but in his popular books he did not take care and thus he ended up misleading a lot of people.
Q1. "One of the proposals mentioned is that the universe has no boundaries or edges in the imaginary time direction. How far as of now is this true? Do we live in open space with no boundaries?"
It's hard to answer this because it's hard to know what you mean by "the imaginary time direction", or what Hawking meant by the phrase. In mathematics it is often useful to introduce the square root of minus one in order to turn one equation into another equation with a sign change; this is a good way to simplify some calculations, or use a previously obtained result. But that does not solve any metaphysical puzzles at all. It shows how the equations of physics can be seen in more than one way; it does not explain why we have those equations and not some other set.
Our understanding of Big Bang cosmology runs up against an early period where energies reach the Planck scale; about that early period we have no certain knowledge, and even rough estimates are almost impossible. Maybe the concept of time itself is no longer valid at some such early stage. But even if we learned more about that, we would still not know how or why those particular components of the cosmos (whatever they may be) came about, as opposed to some other possibility. 
Q2. "Secondly, I want to know whether the universe in which I am living is enclosed or not?"
If you mean of finite total volume (without being bounded) then it is not known but remains a possibility. If the universe has a slight positive curvature overall, then it is finite. If it is flat or negative curvature, then it could be either infinite or finite and with a non-trivial topology. You can take your pick as to which you would be prepared to bet on. I would not stake very much on your bet if I were you.
Q3. "What does the author mean when he says that "we live in a universe that is completely self-contained and without beginning or end.""
I thought about this for a while, and my reaction is like yours: it is hard to know what Hawking meant by this. He seems to be saying that he thought the physical world somehow explains itself, or something like that. But here he is entering into the areas of philosophy called metaphysics and epistemology, and instead of presenting some helpful introduction to the subtleties of those areas, he offers this opinion. The main thing to note is that this opinion of Hawking's is just an educated guess; it cannot be derived from any scientific study because it is meta-scientific. e.g. to say "the universe is without beginning" is to make a meta-scientific or meta-physical claim. To say
"the universe is without end" is equally meta-scientific or meta-physical, because it suggests that some physical model can still be applied in conditions infinitely far removed from the ones where it has been tested, but that is not a scientific thing to do. It doesn't mean one is not allowed to have opinions and hunches on such questions. It does mean one should know that they are just that: opinions, not scientific conclusions.
A: 
One of the proposals mentioned is that the universe has no boundaries or edges in the imaginary time direction. How far as of now is this true? Do we live in open space with no boundaries?

How far as of now is this true? It's true till now as time continues to flow while you are reading this and we have no false evidence to prove that it will not do otherwise in the future. Do we live in open space with no boundaries? Well scientists have long debated over this and till now we don't have concrete experimental evidence to prove that space is bound.

Secondly, I want to know whether the universe in which I am living is enclosed or not?

Look at the above paragraph.

we live in a universe that is completely self-contained and without beginning or end.

This needs a little explanation. Assuming that you have an exposure to basic physics and you have read the book with much enthusiasm, I invite you to think of this. If universe has a beginning what existed before? Did all of the matter, space and time just sprout out in spontaneously? 
If Universe has a end? Where will all the stuff we love go? Do they just vaporise?
To be honest, these questions are very disturbing if not interesting for a logical mind, at least for me. 
Quoting Richard Feynman 

Physicists like to think that all you have to do is say, 'These are the conditions, now what happens next?'   

