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- Prove that every subset of a Hausdorff space is Hausdorff in the subspace topology.

- If
*X*and*Y*are Hausdorff, prove that*X**Y*is Hausdorff.

- Prove or disprove: The image of a Hausdorff space under a continuous map is Hausdorff.

- If ~ is an equivalence relation on a Hausdorff space
*X*, is the space*X*/~ with the identification topology always Hausdorff ?

Solution to question 1 - If
- Show that the topology on
**R**whose basis is the set of half-open intervals [*a*,*b*) is normal. - A
*T*_{1}space is one in which for every pair of points*x**y*there is an open set containing*x*but not*y*.

Prove that a space is*T*_{1}if and only if every singleton set {*x*} is closed.

Prove that the only*T*_{1}topology on a finite set is the discrete topology. - Prove or disprove:

- The product of connected spaces is connected.

- If
*X*is connected, then*X*/~ is connected (where ~ is an equivalence relation).

Solution to question 4 - The product of connected spaces is connected.
- Let
*Y*be the space {*a*,*b*} with the discrete topology. Prove that a space*X*is connected if and only if the only continuous maps from*X*to*Y*are the two constant maps which map the whole of*X*to either*a*or*b*. - If
*A*is connected, prove that the closure*cl*(*A*) is also connected. Deduce that the components of a space are always closed subsets.

Is the interior*int*(*A*) always connected ? - Is the space
**R**with the topology of Question 2 a connected space? What are its components?

What are the components of**R**with:

- the co-finite topology,

- the co-countable topology.

Solution to question 7 - the co-finite topology,
- If
*a*,*b*are points in a topological space, define*a**b*if there is a connected subset of*X*containing*a*and*b*. Prove that is an equivalence relation.

If*a*,*b*are points in a topological space, define*a**b*if there is a path in*X*connecting*a*and*b*. Prove that is an equivalence relation.

Deduce that pathwise connectedness implies connectedness.

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