# Sequentially compact in a topological space?

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?

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Here's Google Books link, in case someone finds it useful. For sequentially compact topological spaces, you might start at Wikipedia. – Martin Sleziak May 6 '12 at 12:47

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