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How is continuity of a function defined using open sets?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In the context of topology, the continuity of a function is defined using the concept of open sets.

A function $f$ mapping a topological space $X$ into a topological space $Y$ is defined as continuous if, for every open set $V$ in $Y$ , the preimage of $V$ —the subset of $X$ consisting of all points $p$ such that $f (p)$ belongs to $V$ —is an open set in $X$ .12

This definition provides a powerful generalization of the traditional notion of continuity found in mathematical analysis, which typically relies on distance-based metrics. By focusing on the relationship between sets rather than specific distances, this topological approach allows for a consistent understanding of continuity across a wide variety of mathematical structures.132