Topology " " for Question 4 is defined by the answer for Question 2 (a). Sorry for the mess.

4. Prove a version of the intermediate value theorem for continuous functions f:XZ, where Z has topology Td defined in Homework 6 Question 2, and X is a connected topological space. The space (Z,Td) is called the digital line: it mimics the topological properties of R but has only countably many points. 2. Let Z be the set of integers, and consider the collection B that consists of all subsets of the form {2n} (one-point subsets given by the even integers) as well as all subsets of the form {2n,2n+1,2n+2},nZ. (a) Sketch these subsets, and check that they form a basis of a topology on Z. Let Td denote this topology