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A topological space is a space studied in topology, the mathematics of the structure of shapes. Roughly, it is a set of things (called points) along with a way to know which things are close together.
More precisely, a topological space has a certain kind of set, called open sets. Open sets are important because they allow one to talk about points near another point, called a neighbourhood of the point. A neighbourhood of a point is simply an open set containing that point. If one did not have the concept of open sets, one cannot define neighbourhoods in a good way. If one tries to define a neighbourhood of a point as any set containing that point, it might just include that point and that point only, not any points near it, or points far away. We also have the concept of closed sets, which are complements of open sets. That is, all of the points not belonging to a certain open sets forms a closed set.
Open sets must follow certain rules so that they match our ideas of nearness. The union of any number of open sets must be open, and the union of a finite number of closed sets must be closed. (The second rule only works for a finite number of closed sets. That is because in many cases a set containing a single point is closed. Any set is made of points. If the second rule applied to an infinite number of closed sets, then every set would be closed.) As a special case, the set containing every point is both open and closed.
A set of points can have many different definitions of what an open set is. One can think of only certain sets as open, or more sets as open. One might even consider every set to be open. The same set with different definitions of open sets form different topological spaces.