When can nets be replaced by sequences? For now, my understanding is very unclear. So please help me.

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*My first question is : Can a sequential space(e.g. a pseudometrizable space) be understood to be a space where any topological property(e.g. open, compact, and so on) can be described in terms of sequences instead of nets? If my question is unclear, then, in other words, when can sequences replace nets?

I see an example in Wikipedia: In sequential spaces, a function is continuous if and only if it is sequentially continuous. So, in this case, we only need to consider sequences, not nets, to determine whether a function is continuous. I am reading so many things containing sequential spaces, uniform spaces, uniform properties to obtain a clear understanding, which is not that successful until now. All wiki or stack exchange posts seem to be perversely dodging my tantalizing question.


*My second question is: An bounded linear operator $T$ acting between two Banach spaces is called compact if for any bounded sequence $\{x_n\}$, the sequence $\{Tx_n\}$ contains a convergent subsequence. In deriving this definition among many other alternative definitions, one use the fact that a metric space is totally bounded if any infinite sequence contains a Cauchy subsequence. In this case, "a sequence" is used instead of "a net", which means that it suffices to consider "sequences" instead of "nets" to determine whether a space is totally bounded. But, total boundedness is not a topological property.(Here, I came to know the concept of "uniform property" and "uniform space". But, these concepts seem to rather interrupt resolving my question.)

Another example I found in Wikipedia is : A metric space is complete if and only if it is sequentially complete. Thus, in this case, one only needs to consider Cauchy sequences, instead of Cauchy nets. Here, completeness is not a topological property. But, we can replace nets with sequences. Is it also due to the fact that any metric space is a sequential space?
I am guessing that the reason for a sequence to be used instead of a net is that the space concerned is a (pseudo-)metric space. I'd like to know the super-clear and simple reason or logic behind this, without being further covered by advanced concepts, which tends to make the issue more complicated. Thanks a lot.
 A: There are not any easy shortcuts or simple general rules here.  Whether you can replace sequences by nets in some situation depends on the details of the situation and how exactly you are using the sequences or nets.  In particular, properties related to compactness tend to be very finnicky about when they can be captured using sequences.  For instance, a metric space is compact iff it is sequentially compact, but this is a quite nontrivial theorem (much harder than, for instance, the theorem that sequences detect continuity of maps on a metric space).  And this theorem does not hold for arbitrary sequential (or even first-countable) spaces: for instance, the ordinal space $\omega_1$ is first-countable and sequentially compact, but is not compact.
As a general rule of thumb, you can always replace nets with sequences when thinking about (pseudo)metric spaces.  I don't know any counterexamples to this, but I also don't know any precise theorem that would cover every possible situation in which you might want to replace nets with sequences.  And once you leave metric spaces to more general spaces like first-countable or sequential spaces, things get much more complicated.
