Show that every Moore space of cardinality less than or equal the continuum has a cou

Hello every body;

My question is

Show that every Moore space of cardinality less than or equal the continuum has a countable separating open cover.

I need this result in order to show that each locally connected, locally compact normal Moore space is metrizable.

I follow the prove I found it in the book " lectures on set theoretic topology" by Rudin but I could not understand it completely. Also, I red the following paper " Metrization of Moore Spaces and generalized Manifolds" by Reed and Zenor

Please help me and I will be very happy

Thank you in advance