You are completely wrong, Duo.
You should to fix it: http://i75.fastpic.ru/big/2016/0115/0c/f94130ab963c1ac2b0f2a0d518426b0c.jpg
You DO NOT have a space, and not I "have an EXTRA space" (c) as you think.
As for the other question: "centrum" is technically the area in the very middle of the city. It's usually the most important part of the city, with shops etc.
"rynek" is harder to explain. It suggests an older part of the city, a representative one, which (hopefully) shows some charm of the past. And obviously has some shops and similar buildings. I think maybe it would be best if you used Google Graphics for that one.
Seems to me that the 'rynek' is the 'market square' or - for Americans, who do not have market squares - 'the old town square' would probably convey the meaning of 'rynek' if you're talking about such a place in a city that has an 'old' and a 'new' town.
I think with 'rynek' you've got to include 'square' in your translation. It's the Polish equivalent to the Marktplazs in Basel, Covent Garden before it was roofed, or something like the Piazza San Marco in Venice or Trafalgar Square in London, or even the 'village green' in English villages and some New England towns, though in Poland, it's definitely connected with an large, open space where people congregated for trade purposes (rather than the space in front of a Church where they would congregate for processions, for example; or a simple 'grand square' meant to show the glory of the culture or commemorate imperial or military triumphs).
The rynek is definitely distinguished from 'the center' by being an open space (formerly an open-air marketplace) ringed with buildings that housed merchants who did business on the ground floor of their private homes. The center is the entire area of the city that includes the rynek but is not limited to it - yes?
"City centre" is a geographical or administrative term that refers to the inner part of the city, which is (in all cases known to me) significantly larger than the so called historic centre aka Old Town. The historic centre is usually the part that already existed in medieval times and which serves as a tourist attraction.
Below you can see the administrative borders of Śródmieście (more or less a synomym for centrum miasta) in Gdańsk (indicated by a thin red line). I tried to approximate the boundaries of the historic centre with an orange circle, which, as you can see is a lot smaller than Śródmieście.