Excluding big bang itself, does spacetime have a boundary? My understanding of big bang cosmology and General Relativity is that both matter and spacetime emerged together (I'm not considering time zero where there was a singularity). 
Does this mean that spacetime (say, away from the big bang event) must have a boundary? 
And what is its topology? By topology I mean without the metric.
 A: Spacetime (probably) does indeed have at least one boundary. Crazy Buddy mentioned three related questions in his comment, and reading these will help you understand why spacetime has a boundary in the past i.e. the Big Bang. This is a singularity and it is a boundary because you cannot follow geodesics back through it to earlier times.
If the universe were closed (the experimental evidence is that it isn't) then there would be a similar boundary in the future i.e. the Big Crunch. This time it would be impossible to follow geodesics forward through the Big Crunch to later times.
In some of the theories of dark energy future boundaries may exist even though the universe is not closed. In particular there is a possible singularity called the Big Rip that would also act as a future boundary. However I should emphasise that these theories are highly speculative.
All the above boundaries are ones that co-moving observers hit by travelling in time. If you're asking if the universe has a boundary in space, i.e. there is a point where some observer could not move in that spatial direction, then as far as we know there are no such boundaries. If the universe is currently infinite then it has always been infinite. Alternatively if it is closed (presumably on some scale much large that the observable universe) then there would by definition be no edge.
On last caveat: I would guess most of us don't believe singularities exist, and some theory of quantum gravity will take over at very short distances and remove the singularity. This would also remove the boundaries I mentioned above. However no such theory of quantum gravity exists at the moment.
Response to comment:
It's a common misconception that the Big Bang was a point where everything came into existance, and therefore that the expanding universe must have an edge because it expanded from a point of finite size. However this is not the case. There are two possibilities:


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*the universe is closed on some very large scale.


In this case consider the analogy of a balloon deflating. A (very small) ant crawling on the balloon would never encounter an edge as the balloon shrinks, so there would be no boundary. The Big Bang is analogous to the point where the ballon shrinks to zero size. What happens at zero size we can't say (because this is a singularity) but at any non-zero size, no matter how small, there is still no boundary.


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*the universe is infinite


This is harder to stretch your brain around. If the universe is infinite then it has always been infinite and remains infinite in size even back to the Big Bang. At the Big bang you get the odd result that the spacing between every point in the universe is zero, but the universe is still infinite. But then this is what makes the Big Bang a singularity. Whatever the case, like the closed universe for any non-zero size, no matter how small, there is still no boundary because the universe is infinite.
A: After the question was originally asked, the OP changed it to exclude the Big Bang. I don't understand the motivation for imagining that there would be a boundary anywhere else. The following answer addresses the question as originally asked.
First, we should recognize that any answer to this question is going to be model-dependent. The Big Bang is the only obvious place to look for a boundary, and the description of the Big Bang is model-dependent. In GR, it's a singularity. This is widely believed to be an indication that GR breaks down there and that we actually need a model of quantum gravity in order to describe the Big Bang. For the remainder of this answer, I'll assume GR as the model.
Within this context, the answer to the question is no: standard cosmological models do not have a boundary at the Big Bang. In terms of mathematical definitions, there are manifolds and manifolds with boundary. It follows directly as an interpretation of the definition of a manifold that all points in the manifold are alike in the sense of how they relate to their immediate surroundings. In a manifold with boundary, you have boundary points that are different, e.g., you can't enclose them with an open set.
The Big Bang is not a boundary. This is because the Big Bang is not a point or set of points in the manifold. In simple terms, imagine a timeline like the ones they use to teach history. In a Big Bang cosmology, the timelike looks like the set defined by $0 < t < \infty$. This set is topologically the same as the entire real line. It has no boundary on the left.

And what is its topology? By topology I mean without the metric.

That's been discussed here: What is known about the topological structure of spacetime?
