Does visible universe have shape of a 3-sphere? Here's my logic:
If you look out in the visible universe you see further back in time. Look enough back and you get to the big bang singularity.
This means whichever way you look in the visible universe you see out to a single point  - the big bang singularity.
What topology is there where you look out in any direction and see the same point? A sphere.
Therefore the visible universe is like a sphere. With you on the north pole and the big bang on the south pole.
Interesting to think about where the equator of this sphere is?
Well, if you look out into space at first you see more and more volume because the surface of a sphere at distance R is proportional to $R^2$, but as you look further back in time until you see a distance where the universe was smaller, so then you see less and less. There must be a certain distance where you stop seeing more and more stuff, but less and less stuff until you get to the big bang singularity itself.
Well, maybe not a sphere, maybe more of a lemon shape.
But my thought is, that if the visible universe is this closed shape... it has no boundary as such.
 A: It's possible that the universe has the form of a curved, spherelike 3d space (you can say curved in a 4d Euclidean space, but in General Relativity, space is inherently curved, inherently curved, hand in hand with time). Like a curved, spherelike 2d sphere, as mentioned in your question.
Observations suggest that the visible universe is flat. So space continues behind the visible universe (behind the horizon). The visible universe could be just a small patch of a huge (immeasurable) curved spherelike 3d space. 
Somewhat like we see the Earth around us. Locally flat, globally curved. On Earth though, an equator is related to the Earth's rotation. To my knowledge, the universe is not rotating (though there are physicists who think that this is the case). So where do you have to put the equator if there is no objective equator?
So to answer your question: the universe could have the form of a very mildly curved spherelike space. For convenience, I didn't include time.
I see, after reading the comments below, that your question is about the visible universe. Well, then you probably can guess the answer after reading what I wrote down above. NO. Which is to say, the 3d space is not a closed space (a balloon in 2d).
You write ** ... it has no boundary as such.** I'm not sure what boundary you refer to. The big bang boundary? Every point in this universe lies on the big bang boundary. Maybe not in an ephyrotic spacetime, but that's still a hypothetical spacetime.
You write:  

This means whichever way you look in the visible universe you see out to a single point - the big bang singularity.

This is just not true, as I already mentioned. Every point in the universe is related to the big bang singularity. 
If you mean a flat 3d sphere, then the answer is obviously yes.
