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The line with two origins is a standard topological counterexample.

Essentially the idea is as follows: imagine the real number line, just like you learned it in high school, but with a second origin (zero) point. In other words, there's another zero, which I will call "z." So every positive number is bigger than z and every negative number is smaller than z, but 0 is not smaller, larger, or equal to z. You might say it's confused with z.

Rigorously, let X be the union of the set of real numbers with an additional singleton z. Take as a basis for a topology on X all real intervals and sets which are the union of the point z with intervals (-a,0) and (0,b) for any positive a and b.

Now, as you might imagine, the line with two origins is very similar to the normal line in most respects. Most importantly, if you shrink in real tight around any point in the line with two origins, you can always get something that looks like a segment of the normal number line. If the point you're shrinking around is positive or negative, you can just shrink in close enough that zero and z are far away.

On the other hand, if the point you're shrinking around is zero, you can just look at everything except z, which is just the number line. Contrarywise, if you're shrinking around z, you can just look at everything except zero, because that also looks like the normal line, with z as the zero point.

Technically, every point in this space has a neighborhood homeomorphic to the real line; this is the main requirement for a space to be a manifold. However, X is not Hausdorff, as any neighborhoods of 0 and z must have non-null intersection. This is pathological for a number of reasons; probably the most convincing is that it prevents X from being given a metric, or distance function. The ability to give a space a metric is a very nice property, and is crucial for most of the geometry that takes place on manifolds. To allow beasts like the line with two origins under the same rubric as "real" manifolds would make the class of object much less useful.

Of course, this generalizes to virtually any topological space you can think of. You might have a circle with a double point on it, the n-dimensional Euclidean space with a repeated point (or line, or space) with a similar topology. The line with two origins is only emphasized because it is the simplest counterexample.

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