An ideal dipole has no gap at all between the two branches.
Reality is that this is technically impossible to realize, and gaps that are very small compared to the wavelength don't matter (enough to care about).
Also note that there's antenna types where the change between the characteristic impedance of the conducted wave on the waveguide (be it a ladder line, a coaxial cable, or something like a rectangular hollow waveguide) and the main emitting elements of a dipole is done exactly by tapering the conductor distance, to give a more wideband match than would be possible with a plain dipole, at the cost of not being as good as a dipole of the same dimension for the lower frequencies. It's trade-offs all the way!
Like the one shown below.
yeah, that's not really a classical dipole to begin with – it would only be if you bent both telescope conductors to lie flat, pointing in opposite directions. And then, that loop in front would still interact with the electromagnetic field, so under no circumstance can this be fully understood as dipole.
The design tradeoffs, probably in order of importance, for such an antenna like that are:
- Marketability – antennas that look fancy are sold easier than antennas that don't, no matter whether the looks contribute to better antenna performance. And, boy, is that a design-intense three pieces of wire!
- Wideband – you'll notice the antenna is sold for "FM and HDTV" reception. That covers the VHF low band TV and broadcast audio broadcast from 44 to 104 MHz, the VHF high-band between 184 and 216 MHz, the lower UHF band from 470 to 614 MHz and from 704 to 890 MHz. No actual dipole could serve all these bands at the same time – a dipole that works well for one frequency can't work well for twice that frequency. So, a proper dipole design is out of the question. Being a good antenna is not what you need to sell an indoor antenna – not being bad at any of these bands is
- Adjustability – this ties in with marketability: because these antennas are inherently bad, and because they are used indoors, interaction with the multipath environment and nearby conductors is what makes and breaks the antenna performance. So, not only do you need to give the user the ability to adjust their antenna in a way such that their piping becomes part of the antenna (totally accidentally!), the older they are, the more likely they are to want that. Back in the day, fewer transmitters at lower frequencies meant that fiddling with wire or telescope antennas was what was needed to get good reception. Hard to explain in marketing material that for higher frequencies, a purpose-designed multi-band antenna, potentially paired with per-band selection combining, would by far outperform their hand-adjusted "living room artistry antenna". Remember that as antenna manufacturer, you don't live off the educational material on wave propagation and antenna theory tha you sell, but off people buying an antenna to be able to see their TV series or animal-named news broadcaster.
So, very little of the antenna and wave theory education in this world applies to the commercial antenna you show. Antenna theory can't justify mounting a loop directly next to two adjustable branches of some kind of "bent" dipole; the answer to your question hence is "the gap is as large as looks stylish, and still works when tested in a real living room".