Prove $[0,1)$ is of cardinality equal to $(0,1]$

Question

Let $A=[0,1), B=(0,1], C=(0,1)$.

Prove $A$ is of cardinality equal to $B$ by using Glueing Lemma to 'glue together' the identity function id$_C$ and some appropriate bijective function.

I'd like to ask what does it mean by 'glueing identity function and some bijective function'? How to find those 'appropriate bijective function'?

Thanks.

Answer 1

The glueing lemma for two maps:

Lemma. If $A = A_1 \cup A_2$ and $B = B_1\cup B_2$ are decompositions (that is $A_1\cap A_2 = B_1\cap B_2 = \emptyset$), and $f_i \colon A_i \to B_i$ are bijectons for $ i = 1,2$, then $f = f_1\cup f_2$ is a bijection $A\to B$.

Here $A_1 = B_1= C$ and $ f_1= \mathrm{id}_C$ ($L$ should be a typo). Can you find $A_2 = A-A_1$ and $B_2 = B-B_2$? Are these two sets equinumerous?