The cardinality of Borel set and real number

Question

My understanding of Borel set is the collection of all open/close intervals on $\mathbb{R}$. So if we create a map from Borel set to $\mathbb{R}$, namely map all the open/close intervals centred at the same real number to that number in $\mathbb{R}$, we can see that it is a many-to-one mapping. Therefore, the cardinality of Borel cannot be smaller than that of $\mathbb{R}$. However, a measure theory lecturer says the cardinality of Borel is less than or equal to $\mathbb{R}$, depending on the continuum hypothesis. Could someone help to clarify?