What is a topology?

For this lecture, replace the four-card board with a separate carrier containing just two points, zero and one. A topology specifies which subsets count as open. It is a collection of subsets of this carrier, rather than one distinguished subset or a geometric arrangement of the two points. Consider the empty set, the singleton containing one, and the whole two-point set. We propose exactly these three opens. The singleton containing zero is absent. To check the proposal, we must test the topology rules. For the set notation, braces list members and the empty-set symbol means no members. Union collects members from either set. Intersection keeps only shared members. We are working on the separate two-point carrier, zero and one. The empty set and the whole carrier must be open. Our list includes both. Any union of listed opens must also be listed; unions in this example give only the empty set, the singleton containing one, or the whole carrier. Any finite intersection of listed opens must stay on the list. Intersecting the whole carrier with the singleton leaves that singleton; intersecting with the empty set gives the empty set. Repeating a set adds nothing, so the proposed family passes. Arbitrary unions allow any collection of open sets, including an infinite collection in a more general space. Finite intersections require only finitely many opens at once. These are different requirements, even though our current carrier has only finitely many subsets. On the real line, the usual open sets are unions of open balls. Small balls around a point fit inside such an open set. That familiar construction motivates topology, but a topology can also be specified directly, as our two-point example shows. We will use open neighborhood to mean an open set containing the point in question. The singleton containing one is an open neighborhood of one. The only open neighborhood of zero in our example is the whole carrier. An open neighborhood of one can exclude zero, but no open neighborhood of zero excludes one. Our definition permits this asymmetry. A topology is not required to make every singleton open or to come from an ordinary metric. A discrete topology declares every subset open. On two points it has four opens. An indiscrete topology declares only the empty set and the whole carrier open. Our three-open example lies between those choices. A set is closed when its complement is open. In the three-open example, the singleton containing zero is closed because its complement is the open singleton containing one. Closed does not mean empty of information or physically bounded by a line. The empty set and the whole carrier are both open and closed, or clopen. In a discrete topology every subset is clopen. A set can therefore be both open and closed; these words are not exclusive alternatives. Must the singleton containing zero be open because the singleton containing one is open? Check the topology rules, rather than appealing to symmetry in the picture. No. The three listed sets already satisfy the rules, and neither their unions nor their finite intersections produce the missing singleton. Taking complements is not a requirement for the collection of open sets. We have a legal topology with asymmetric neighborhoods. Next we will construct such families from selected subsets instead of being handed a finished list. Keep the generating subsets separate from the completed collection of opens.