What is the definition of topological space? Explain the concept and show an example.

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A topological space is a fundamental concept in mathematics that generalizes the notion of open sets. Formally, a topological space consists of a set X together with a collection tau of subsets of X, called a topology, that satisfies three axioms. First, both the empty set and the entire set X must belong to tau. Second, the union of any collection of sets in tau must also be in tau. Third, the intersection of any finite collection of sets in tau must be in tau. The sets in tau are called open sets. Let's look at a concrete example of a topological space. Consider the set X consisting of three elements: a, b, and c. We define a topology tau on X as the collection of subsets: the empty set, the singleton set containing just a, the set containing a and b, and the entire set X. To verify this is indeed a topology, we need to check the three axioms. First, both the empty set and X are in tau. Second, any union of sets in tau is also in tau. For example, the union of the singleton set a and the set containing a and b gives us the set a,b, which is in tau. Third, any finite intersection of sets in tau is in tau. For instance, the intersection of the set a,b and the entire set X is just a,b, which is in tau. Since all three axioms are satisfied, the pair X,tau forms a topological space. Let's explore some common types of topologies. The discrete topology on a set X consists of all possible subsets of X. In this topology, every subset is considered open, making it the finest possible topology. At the other extreme, we have the indiscrete topology, which contains only the empty set and the entire set X. This is the coarsest possible topology. A familiar example is the standard topology on the real line, where open sets are defined as unions of open intervals. This is the topology most commonly used in calculus and analysis. These examples illustrate how the same set can have different topological structures depending on which collection of subsets we designate as open. A fundamental concept in topology is that of continuous functions. A function f from a topological space X to a topological space Y is continuous if for every open set V in Y, its preimage under f is an open set in X. This generalizes the familiar notion of continuity from calculus. For functions between real numbers with the standard topology, this topological definition is equivalent to the epsilon-delta definition you may have seen in calculus. Continuous functions have several important properties: the composition of continuous functions is continuous, the identity function is continuous, and constant functions are always continuous. This definition of continuity captures the intuitive idea that continuous functions preserve the 'nearness' of points - points that are close in X map to points that are close in Y. To summarize what we've learned about topological spaces: A topological space consists of a set X together with a collection tau of subsets that satisfy three key axioms regarding the empty set, unions, and intersections. This collection defines which subsets are considered 'open,' generalizing the familiar concept of open sets from metric spaces like the real line. We've seen several examples of topologies, including the discrete topology where every subset is open, the indiscrete topology where only the empty set and the whole space are open, and the standard topology on the real line. Continuous functions between topological spaces are those that preserve the open set structure through preimages. The power of topology lies in its ability to study properties that depend only on the 'nearness' of points, without requiring a notion of distance. This makes topology a fundamental tool in modern mathematics, with applications ranging from analysis and geometry to data science and theoretical physics.