This question is related to topological manifolds, as discussed in one of Frederic Schuller's lectures on gravity : https://www.youtube.com/watch?v=93f-ayezCqE.
A bit of background: Briefly, to study spacetime, we assign it the structure of a topological manifold $M$ because one can then map its open subsets to open subsets of $\mathbb{R}^d$, via chart maps. One can then infer the continuity of the "real" curve in spacetime (say, $\gamma : \mathbb{R} \to M$) from the continuity of the curve (that it's mapped to) in $\mathbb{R}^d$. More generally, one can infer the properties of spacetime by looking at the properties of its chart maps (atlas).
The lecturer stresses that while the curve in the manifold is a "physically real thing", the curve in $\mathbb{R}^d$ isn't, since it depends on our choice of the chart map, which in turn can be arbitrary. Therefore, we are concerned with those properties of $\mathbb{R}^d$ curves that are independent of the choice of the chart map, and we can only talk about such properties being applicable to $\gamma$. Continuity happens to be a chart-independent property and so we can talk about "continuity of $\gamma$".
Question: Doesn't the notion of the continuity of spacetime curves itself depend on our choice of the topology on $M$? Does that not rule out continuity as a "physically real property" of a "physically real" curve, since it's not independent of the choice of topology? If that is indeed the case, why go through all the effort of inferring the continuity of $\gamma$, instead of being satisfied with the continuity of chart map ($\mathbb{R}^d$) curves?