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

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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: