The Essence of Topology

Topology, according to Wikipedia, is the branch of mathematics concerned wth the properties of a geometric object that are perserved under continuous deformations, such as stretching, twisting, crumping, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. However, I personally find this definition misguiding in many ways.

The definition puts fowards many different forms of continuous deformations, making it seem like these transformations are necessary for understanding topology. It makes topology seem like something only relevant where such transformations actually exist, which is not at all true. Topology, in its essence, is a study of structure. In set theory, we investigate the most elementary of characteristics of sets which is it’s size, i.e., its cardinality. However, in topology, we go further, endowing sets the notion of nearness, compactness, and connectedness.

Nearness is provided to a set using open sets. By putting elements into an open set, we are arbitraily defining the points to be near each other. With this intuition, the definition of continuity naturally follows, by saying that a points nearby each other in one set would be mapped onto points near each other on the other. Note our use of open sets rather than closed. This comes from the fact that operations such as differentiation is done on open sets, requiring that we have nearby points for all points that are relevant.

Compactness is the notion of having no “punctures” or “missing endpoints”. The concept sheds light to a more global perspective of a set.