It is known that the Cantor set minus a point is (up to homeomorphisms) the unique zero-dimensional separable metric space without isolated points that is locally compact and not compact. In particular, the Cantor set minus a point is homeomorphic to the Cantor set minus two points (or minus any finite number of points).

Now you can take as $X$ the Cantor set and as $Y$ the Cantor set minus a point.

See also this math.stackexchange question.

It is known that the Cantor set minus a point is (up to homeomorphisms) the unique zero-dimensional separable metric space without isolated points that is locally compact and not compact. In particular, the Cantor set minus a point is homeomorphic to the Cantor set minus two points (or minus any finite number of points).

Now you can take as $X$ the Cantor set and as $Y$ the Cantor set minus a point.

See also this math.stackexchange question.

It is known that the Cantor set minus a point is (up to homeomorphisms) the unique zero-dimensional separable metric space without isolated points that is locally compact and not compact.

So take as $X$ the Cantor set and as $Y$ the Cantor set minus a point.

See also this math.stackexchange question.

It is known that the Cantor set minus a point is (up to homeomorphisms) the unique zero-dimensional separable metric space without isolated points that is locally compact and not compact. In particular, the Cantor set minus a point is homeomorphic to the Cantor set minus two points (or minus any finite number of points).

Now you can take as $X$ the Cantor set and as $Y$ the Cantor set minus a point.

See also this math.stackexchange question.