# Counterexamples around connected spaces

A connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. We look here at unions and intersections of connected spaces.

### Union of connected spaces

The union of two connected spaces $$A$$ and $$B$$ might not be connected “as shown” by two disconnected open disks on the plane.

However if the intersection $$A \cap B$$ is not empty then $$A \cup B$$ is connected.

### Intersection of connected spaces

The intersection of two connected spaces $$A$$ and $$B$$ might also not be connected. An example is provided in the plane $$\mathbb R^2$$ by taking for $$A$$ the circle centered at the origin with radius equal to $$1$$ and for $$B$$ the segment $$\{(x,0) \ : \ x \in [-1,1]\}$$. The intersection $$A \cap B = \{(-1,0),(1,0)\}$$ is the union of two points which is not connected.

# A connected not locally connected space

In this article, I will describe a subset of the plane that is a connected space while not locally connected nor path connected.

Let’s consider the plane $$\mathbb{R}^2$$ and the two subspaces:
$A = \bigcup_{n \ge 1} [(0,0),(1,\frac{1}{n})] \text{ and } B = A \cup (\frac{1}{2},1]$ Where a segment noted $$|a,b|$$ stands for the plane segment $$|(a,0),(b,0)|$$. Continue reading A connected not locally connected space