explain the definition of connectedness in topology, and explain the intuition
user
Connectedness in topology refers to the notion that a space is a single connected piece, rather than being composed of multiple pieces. Intuitively, this means that if you draw a line in the space, you can draw it so that it doesn't leave any gaps or holes, and it passes through every point in the space. Formally, connectedness is defined as a space that cannot be written as the union of two disjoint non-empty open sets. This means that if you take two non-empty open sets in the space, they cannot have no points in common, and therefore cannot be separated by a line. This definition captures the intuition that a space is either connected or not, rather than having varying degrees of connectedness.
assistant
Can you please make it easier to understand?
user
Roughly speaking, a connected topological space is one that is “in one piece”. The way we
will define this is by giving a very concrete notion of what it means for a space to be “in two or
more pieces”, and then say a space is connected when this is not the case.
assistant