Topological Spaces and Continuous Functions
• Topology, open set, neighborhood, closed set
• Closure, interior, boundary, limit point, Hausdorff topology, separable space
• Base, countable base, subspace topology, order topology, metric topology, finite product topology
• Continuous mapping, open map, closed, map, quotient topology, identification map

Connectedness
• Connected space, continuous image of a connected space, path connected space, continuous image of a path connected space
• Product of two path connected spaces, product of two connected spaces
• Locally connected space, locally path connected space, component, path component

Compactness
• Compact spaces, tube lemma, finite intersection property, closed and bounded subsets of Euclidean spaces, Cantor sets
• Limit point compactness, Lebesgue number of a covering, local compactness, one-point compactification
• Continuous image of a compact space, the product of two compact spaces, uniform continuity and compactness

References:

Topology, Sheldon Davis, McGraw-Hill.
Topology: A First Course, James R. Munkres, Prentice-Hall.