Previous page (Exercises 5) | Contents | Next page (Exercises 7) |

- If
*U**A**X*, prove that*U*is closed in the subspace topology on*A*if and only if*U*=*A**Z*for*Z*a closed subset of*X*.

Prove or disprove the following:

- The interior of
*U*in the subspace topology on*A*is equal to the interior of*U*in the topology on*X*,

- The closure of
*U*in the subspace topology on*A*is equal to the closure of*U*in the topology on*X*.

Solution to question 1 - The interior of
- Let
_{1}and_{2}be the subsets of the natural numbers**N**defined by:

*U*_{1}if either*U*= or**N**-*U*is finite,

*U*_{2}if either 1*U*or**N**-*U*is finite.

Prove that_{1}and_{2}are topologies on**N**.

Let*f*be the identity map on**N**and let*g*be the map from**N**to**N**defined by

*g*(*n*) = 1 if*n*is odd;*g*(*n*) = 1 +*n*/2 if*n*is even.

Determine whether*f*and*g*are continuous either as maps from (**N**,_{1}) to (**N**,_{2}) or as maps from (**N**,_{2}) to (**N**,_{1}). - Let
*A**Y*and*B**Y*so that*A**B**X**Y*. Prove that:

*cl*(*A*)*cl*(*B*) =*cl*(*A**B*)

*int*(*A*)*int*(*B*) =*int*(*A**B*)

*cl*denotes the closure and*int*denotes the interior. - A set of subsets of a topological space
*X*is called a*sub-basis for the topology*if every open set can be written as a arbitrary union of finite intersections of sets in .

Show that a function*f*from a topological space*X*to a topological space*Y*is continuous if and only if*f*^{-1}(*U*) is open for every set*U*in a sub-basis for the topology on*Y*.

Prove that the set of all unbounded open intervals of**R**forms a sub-basis for the usual topology on**R**which is not a basis.

Prove that a sub-basis of the product topology on*X**Y*is the set of subsets of the form*U**Y*and*X**V*for*U*_{X}and*V*_{Y}. - Prove that the set of all -neighbourhoods of rational points of
**R**with also rational, forms a basis for the usual topology on**R**. Deduce that the usual topology on**R**has a countable basis. Prove that the discrete topology on**R**does not have a countable basis. - Let
*X*be the open unit square and let*Y*be the open unit quadrant, each with their topology as subsets of**R**^{2}. Prove that the map which takes the point (*x*,*y*) (in Cartesian co-ordinates) to (*x*,^{1}/_{2}*p**y*) (in polar co-ordinates) is a homeomorphism.

Prove that this homeomorphism cannot be extended to a homeomorphism between the*closed*unit square and*closed*unit quadrant.

Show, however, that the closed unit square and closed unit quadrant*are*homeomorphic. - Prove that [0, 1) (0, 1) and [0, 1) [0, 1] are homeomorphic subspaces of
**R**^{2}.

Previous page (Exercises 5) | Contents | Next page (Exercises 7) |