Metaphor of topological basis and open set

The definition of topological basis for a space $X$ requires that each point $x$ in $X$ is contained in one of the said topological bases. Meanwhile, the definition of open set also claims that a topological basis can be drawn around each point of the set. Because the topological basis itself is also an open set, such operation of drawing neighborhood around each point can be continued forever.

The abstract concept on topological basis and open set can be likened to throwing pebbles into a water pond without physical boundary, so that no matter where a pebble is dropped into the pond, there will be ripples produced, which are embodied as concentric rings. Then, an open set is similar to such a pond without boundary and ripples in the form of concentric rings are just those local topological bases drawn around the pebble’s incidence point. An example of such concentric rings is the usually adopted topological bases for $\mathbb{R}^n$ which are defined as open balls.

Another point to be noted is since the concept of metric has not come into play yet, the said “without boundary” implies some kind of infinity, which does not mean a distance with infinite length, but drawing local topological bases one inside another ad infinitum. In addition, because such kind of never-stopping self-nesting operation can be applied to each point in an open set, some sense of continuity can bud from here. And it is natural to see that the definition of continuity relies on open set.

Fig. Throwing pebbles into a pond as the metaphor of topological basis and open set.

巴特西

Metaphor of topological basis and open set

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