My textbook 'Introduction to Metric and Topological Spaces' goes though the definition of a compact space in a metric and topological space, and also sequentially compact spaces in a metric space.
My question is, can a topological space be sequentially compact? And if so is there any sort of equivalence between sequentially compact and just compact like there is in a metric space?
A topological space can be sequentially compact, but it is not an equivalent property to compactness in general as it is in metric spaces. There are counter-examples of non-compact spaces being sequential compact, and vice versa.
If you're interested on the equivalence of different types of compactness I suggest you to look up a property called 'first countability'. A space is first countable $(N_{1})$, if every point has a countable neighbourhood basis. Without this property, sequences are quite powerless in general topological spaces and one is usually forced to work with nets or filters instead. In fact, one can show that sequential compactness is equivalent with countable compactness (i.e. every countable cover possessing finite subcover) in a $N_{1}$ space.
A long time ago while giving a course in analysis I noticed how elegantly some of the work turned out if one concentrated on the use of sequences and the notion of sequentially compact. This led me to look at the notion of a one-point sequential compactification: add an extra point to which all the non convergent sequences in $X$ should now converge! This led to the paper
Brown, R. "Sequentially proper maps and a sequential compactification'', J. London Math Soc. (2) 7 (1973) 515-522.
