Continuity on Subsets of Topological Spaces

In general topology, a function $f:T \rightarrow U$ is defined to be continuous if the preimages of open sets in $U$ are open in $T$ . This doesn’t define however functions that are continuous on just a subset of $T$ or at even a point. An application of continuity on subsets would be topological fields where the multiplicative inverse couldn’t possibly be continuous on all of $T$ , since $T$ includes 0. Since topological fields are standard and well-known concepts, the idea of generalized continuity on subsets must also be standard and well-known. In this article, I scour the Internet to understand how we define this today according to already-established standard literature.

As the link https://en.wikipedia.org/wiki/Continuous_function#Continuity_at_a_point states, we can generalize the metric space continuity definition (which works well for subsets of $T$ ) directly to arbitrary topological spaces with the definition given there; this definition is thus equivalent to the metric space definition when $T$ is a metric space. Continuity at a point allows defining continuity on subsets, and this seems like the standard approach.

One question I have: I can think of another “reasonable” definition, that a function $f$ is continuous on a subset $S \subseteq T$ if the preimage of an open set in $U$ is only required to be open if it is within $S$ . In other words: $f$ is continuous on $S$ if for any open set $V \subseteq U$ , if the preimage of $V$ is a subset of $S$ , then this preimage is open. Is this definition equivalent to the one above? Would it also be equivalent if we consider a point to be a one-element subset? If not equivalent to the one above, is it separately equivalent to the metric space definition for $T$ a metric space? There are some questions to explore here!

First written some time ago (not sure when.)

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EDIT (October 27, 2022): I now know that a generally used definition of continuity on subsets is via the subspace topology; this is how continuity is defined for the multiplicative inverse in topological fields. Thus, we can think of any function $f:T \rightarrow U$ with a subset $S \subseteq T$ as a function $f\left. \right|_{S}:S \rightarrow U$ where $S$ is a topological space in its own right, “ignoring the rest of $T$ .” Then, continuity of $f$ on the subset $S$ is just the same as continuity of $f\left. \right|_{S}$ on its whole domain. Thus, as the Wikipedia article implies, continuity on a subset is equivalent to continuity at every point in the subset, since continuity on the whole domain is equivalent to continuity at every point in the domain.

I’m not sure if this is equivalent to the last definition I came up with; it seems that since I’m relaxing some conditions in the definition (only requiring the preimage to be open if it’s within $S$ ) it will turn out to not be equivalent and be a looser condition. It seems we could come up with a counterexample to demonstrate the lack of equivalence.

It remains to figure this out.

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