As long as politics is not one-dimensional, there are completely reasonable cases where you get a rock-paper-scissors situation between 3 top candidates if voters select whomever is closest to them. That in turn violates the criteria that says the election result can't change if you add a non-winning candiate (rock beats scissors, but when paper enters the race now scissors is computed the winner).
This criteria is called "independence of irrelevant alternatives". A common criticism of Arrow's theorems usefulness is that it is a bit of a stretch to call paper an "irrelevant alternative" when rock-paper-scissors forms a cycle of preferences like that.
If politics is one-dimensional ("single-peaked preferences" in the article), then you never get this situation.
It's also important to note that a much more reasonable criteria exists called "local independence of irrelevant alternatives". This is the idea that total losers joining the race don't affect the result, however someone who is in the top rock-paper-scissors cycle (the "Smith Set") can still affect the result even if they don't win themselves. This is far more reasonable, as it's fairly arbitrary which of the candidates in that top cycle should win.
When there is no top cycle, the Smith Set is a single person (the "Condorcet Winner"). When there is a Smith Set, most reasonable voting systems will pick their winner somewhat arbitrarily as a member of the Smith Set, since everyone in the Smith Set beats everyone outside the Smith Set.
This criteria is called "independence of irrelevant alternatives". A common criticism of Arrow's theorems usefulness is that it is a bit of a stretch to call paper an "irrelevant alternative" when rock-paper-scissors forms a cycle of preferences like that.
If politics is one-dimensional ("single-peaked preferences" in the article), then you never get this situation.
It's also important to note that a much more reasonable criteria exists called "local independence of irrelevant alternatives". This is the idea that total losers joining the race don't affect the result, however someone who is in the top rock-paper-scissors cycle (the "Smith Set") can still affect the result even if they don't win themselves. This is far more reasonable, as it's fairly arbitrary which of the candidates in that top cycle should win.
When there is no top cycle, the Smith Set is a single person (the "Condorcet Winner"). When there is a Smith Set, most reasonable voting systems will pick their winner somewhat arbitrarily as a member of the Smith Set, since everyone in the Smith Set beats everyone outside the Smith Set.