What is a topological space?
Q: What is a topological space?
A: A topological space is a set of points along with a way to know which things are close together. It is studied in the mathematics of the structure of shapes.
Q: What are open sets?
A: Open sets are important because they allow one to talk about points near another point, called a neighbourhood of the point. They are defined as certain kinds of sets that can be used to define neighbourhoods in a good way.
Q: What must open sets follow?
A: 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.
Q: What is the special case for open and closed sets?
A: The special case for both open and closed sets is that the set containing every point is both open and closed, as well as the set containing no points being both open and closed.
Q: How do different definitions affect topological spaces?
A: Different definitions for what an open set can affect topological spaces by considering only certain sets as open or more than usual, or even considering every set to be open.
Q: Can infinite numbers of closed sets form any set?
A: No, if infinite numbers of closed sets were allowed then every set would be considered closed since any set consists only out of points.
