Set-theoretic topology introduces many mathematical concepts simplifying the concepts from other mathematical disciplines, including analysis, graph theory, knot theory, and geometry. The simplification results from liberating parts of the terminology of these other disciplines (including terms such as point, closed intervals, similarity, continuity) from specific terms that seem to exist only in these disciplines on their own.
The notion of a continuous function $f,$ as an example, is explained in the analysis using the epsilon-delta definition. Basically, it states that two points $f(x),f(a)$ get arbitrarily "close" to each other, if $x,a$ are sufficiently close to each other. If the concept of "closeness" is not available, continuity can also be explained using so-called open sets. For instance, in the set of real numbers, open intervals (which are another example of a term from another discipline, which can is simplified in the set-theoretic topology) are open sets from a set-theoretic topological point of view. Thus, open sets can be defined for any other kinds of sets, not only real numbers.
In this part of BookofProofs, we will deal with topological spaces, i.e. sets with open subsets. The general topological spaces get new useful properties if we require them to have additional properties. One such interesting property will be the Hausdorff property: We will require that any two distinct points in the space can be put into disjoint open sets containing them respectively.
A continuous function $f$ that has a continuous inverse $f^{-1}$ is known as homeomorphism. Two topological spaces $X$ and $Y$, between which there is a homomorphism, then one can be deformed into another smoothly enough to consider them equivalent. An interesting topological problem is to classify of spaces which can be deformed into each other smoothly using a homeomorphism.
Another useful feature of topological spaces is their basis $\mathcal B$, which is subset of its open sets such that all other open sets can be described as a union of the basis elements.
