Topology

Open and Closed Sets: Open sets are subsets of a space that are defined to be ‘open’ under some notion of openness. Closed sets, on the other hand, are defined to be complements of open sets, that is, a subset is closed if and only if its complement is open.

Continuity and Homeomorphisms: Continuity is the primary concept in topology, it is defined as a function between two topological spaces that preserves their underlying topological structure. A homeomorphism is a bijective continuous function, its inverse is also continuous. Two spaces are homeomorphic if they can be transformed into each other through a homeomorphism.

Compactness: A topological space is compact if every open cover of it has a finite subcover. It's a fundamental property that is used in many areas of mathematics.

Connectedness: A topological space is connected if it cannot be separated into two non-empty, disjoint open subsets. This concept is used to describe the topological properties of spaces.

Metric Spaces: A metric space is a set of points with a distance function satisfying certain properties. It is important in topology as it is a generalization of Euclidean space.

Separation Axioms: Separation axioms are properties of topological spaces that describe how to separate points and subsets from each other. Separation axioms include T0, T1, T2, T3, and T4.

Topological Groups: A topological group is a group equipped with a topology that is compatible with the group operations. This concept is frequently used in Lie theory, functional analysis, and geometry.