Tile over/around french drain

Take a look at these photos of once corner of my basement laundry room:
http://www.704westjoppa.com/french1.gif
http://www.704westjoppa.com/french2.gif
It appears that the previous owners may have put in a partial french drain here. My issue is that I am re-tiling this floor in commercial vinyl tile and I want to know if tI should cover this area over partially? I know that I should not cover the whole thing--if water should come down the wall (which it never has in the year we have been in the house) I know it needs to get to the french drain.
If I don't cover part of the opening, though, there will be a gap that shows, even after I put up new moulding.
Any problems with covering part of the opening, just so a gap does not show?
Thanks for your thoughts!
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a floating slab. Do not fill this in.
Also - It looks like you do have a water problem, not severe, but you have had at least moisture in the past. What ever that wall treatment is you can see where it has puffed out. This is definitely from water. Its obvious that, one, your wall treatment is not suitable for moist areas, and two, that you do have moisture.
If you do tile, I would say you would be safe tiling over the gap but just to where the baseboard will cover the tile. Do not go tight to the wall. I would also take preventative measures against future water. Seal the interior walls with dryloc and then fix the exterior grade. I'll bet you a dollar that on that side of the house there is an area where you do not have positive slope of grade away from the house or if you do you have a downspout dumping close to the foundation. Correct the grade, extend your downspouts and seal the walls. I would not finish a basement w/o at least doing these things.
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