1 · Compare two ideas of closeness

The two rows contain the same real numbers. In the top row, every point stands alone. In the bottom row, the numbers have their usual distance, so nearby values remain close.

Try this. Make the viewing window smaller. The discrete row eventually isolates one point. An interval in the usual real line always contains more points, however small the interval becomes.

The identity map has zero point-level kernel and cokernel, but it is not an isomorphism of topological groups. Example 1.9 shows how condensed groups keep the quotient that the pointwise test misses.