When we say that the $L^p$ space is complete and hence closed, what ambience space is it closed in, and what is the topology of that ambience space? I think it's the space of measurable functions, but is there a natural norm/topology on the space of measurable functions?
An Update After Reading the Comments:
I realized that this might just be a simple topology question rather than $L^p$ spaces. This is what I have gathered: If we view $L^p$ itself as the ambience space with the $L^p$ norm topology, then of course it is closed (closed and open at the same time) in itself. Completeness has nothing to do with closedness of the space in this case. On the other hand, if we can embed $L^p$ into any topological space $(X, \mathcal{T})$ and if we can show the $L^p$ norm topology is the same as the induced subspace topology of $L^p \subseteq X$, then $(L^p, \| \cdot \|_{L^p}) \subseteq (X, \mathcal{T})$ must be closed in $(X, \mathcal{T})$ as it is complete.
I suppose a question that I have not been able to figure out still is : We know $L^p$ must be embedded in the space of measurable functions. Is there a natural topology that we can put on the space of measurable functions, so that the induced subspace topology of $L^p$ is exactly the same topology as the $L^p$ norm, without knowing if the function domain is finite measure or not?