CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Algebraic topology is generally the study of functors from nice categories of spaces to algebraic categories. This can be considered, from the nPOV, as closely related to higher category theory. Category theory originally developed out of algebraic topology, where it was used first simply to describe what was going on and then to axiomatise Eilenberg-Steenrod cohomology theories. There has been considerable convergence of the two subjects, not only in methodology, but in aims and motivations.
A textbook with an emphasis on homotopy theory is in
Lecture notes include
Brief indications of open questions and future directions (as of 2013) of algebraic topology and stable homotopy theory are in
Tyler Lawson, The future, Talbot lectures 2013 (pdf)
Davis, Lecture notes in algebraic topology (pdf)
Further online resources include