Covering Space Examples - The Bouquet of Two Circles

# Covering Space Examples - The Bouquet of Two Circles

Recall from the Covering Spaces page that if $X$ is a topological space then a covering space of $X$ is a pair $(\tilde{X}, p)$ where $\tilde{X}$ is a path connected and locally path connected topological space and $p : \tilde{X} \to X$ is a continuous map such that for every $x \in X$ there exists a path connected open neighbourhood $U$ of $x$ (called an elementary neighbourhood of $x$) such that $p$ restricted to every path component of $p^{-1}(U)$ is a homeomorphism onto $U$.

We will now look at some examples of covering spaces of the bouquet of two circles

Recall that the bouquet of two circles is the topological space obtained from by attaching two circles at a single point. We label the edges $a$ and $b$ and give an orientation for these edges:

An example of a covering space of the bouquet of two circles is illustrated below:

If we take an open neighbourhood around any point in the bouquet of two circles, then we obtain infinitely many open neighbourhoods in this covering space.

Another example of a covering space of the bouquet of two circles is: