In metric spaces finite intersections of open subsets, all unions of open subsets, the whole space and the empty set are always open. A topological space is a non-empty set with a collection of subsets, which possesses those properties.
Intuitively, we say that a function f is continuous at a particular element b, if images f(x) may be made as close as we like to the image f(b), for all x sufficiently close to b. The topological interpretation is: f is continuous at b if, for every open subset H containing f(b), we can always find an open subset G, whose image, f(G) is contained in H.
About the Author
I had a thorough training in analysis and general topology as an undergraduate, and my doctoral research was in nets and compactifications. I have taught analysis and general topology in three universities, most recently in Asia, but earlier in the UK Open University, where I was also involved in mathematics education. I enjoy helping students appreciate the precision and rigour of analysis and topology, and I continue to gain much pleasure through investigating their secrets.