OK, maybe a silly question but what does this really mean in terms of self-defense? (Since this is a self-defense form, after all). By that I mean, closing the gap is, I asume, a movemtn in to an attack which I wonder if is advisable. There are, I believe, legal ramifications if you are seen as the attacker. I suppose if you are in a 'street fight' and you just want to beat the other guy up then it makes sense, but I'm not sure if it's a good idea for a martial artist defending himself to initiate the action. So I admit that in a self-defense context, I can't envision a context where this is really a good idea, morally or legally.
Nevertheless, assuming that it's a good idea. I think it really depends on your art, or at least the techniques you want to attack with. The 'gap' is going to be different between someone who wants to kick (big gap) someone who wants topunch (medium gap) and somone who wants to grapple (small gap). Nothing wrong with any of them, just different approches. What might seem 'outside the gap' for someone may seem 'pretty close, within striking distance' for someone else. If you are facing someone who likes to kick, you might be thinking you need to close the gap only to find out that you have already crossed that gap, to them, and are about to get struck.
Anyway, how I usually see the gab crossed, from the perspective of an art that wants to kick a lot, is a) deception and b) foot work. Deception is simply to make a motion that simulates an attack to get the opponent to react to the fake but then to come through with an attack from a different attack or angle. For example, assuming you are in a 'fighting stance', you can drive the back leg forward like you are going to kick, but then roll your hips over and kick from the other leg. I know in a different thread bout jabs it was mentioned faking with a jab and coming through with a reverse punch. Same idea, come in with a fake attack to get them to react and use this forward motion to bring you into range for the real attack. I know in another line of thinking there was the idea that is you are going to shoot in for a takedown, you start out by faking a hand attack up to distract and unbalance the opponet. Either way, how you fake depends on what you really want to attack with. The other is footwork. I know from TKD that many kick attacks are intiated with foot movement that brings you into range for the attack. It can be as simple as a slight shuffle forward. One thing I like to do on spinning kicks is to drive the lad foot forward a few inches and turn it as it helps get me rotating and also brings me close enough to be in range. A switch stance with a slight forward motion can initiate an attack.
I guess just thinking of TKD there are too many ways to describe of using footwork to bring you in closer as you start an attack so...each art is going to have their own ways of coming in close as part of an attack technique so...
...it really depends on how you are trying to attack