Let $ \ (X, d) \ $ be a metric space and let $ \ A ⊆ X \ $. Suppose that $ \ (A, d) \ $ is homeomorphic to a complete metric space $ \ (Y, d_Y ) \ $. Prove that $ \ A \ $ is a $ \ G_\delta \ $ subset of $ \ X \ $.
Answer:
Since $ \ (A,d ) \ $ is homeomorphic to a complete space , then $ (Y,d_Y) \ $ can be embedded into $ \ X \ $ .
Now since $ A \ $ is complete , we can write $ A=\cup_{n \in \mathbb{N}} \ C_n \ $ , where $ C_n \ $ are closed set .
But now can not conclude the proof .
please help me out
