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- Prove that in a discrete metric space, every subset is both open and closed.

If*f*is a map from a discrete metric space to any metric space, prove that*f*is continuous.

Which maps from**R**(with its usual metric) to a discrete metric space are continuous ? - If
*f*from**R**to**R**is a continuous map, is the image of an open set always open ?

Is the inverse image of a closed set always closed ? - Show that in any metric space an -neighbourhood is an open set.

Show that any open set can be written as a union of suitable -neighbourhoods.

Give an example of an open subset of**R**(with its usual metric) which cannot be written as a union of finitely many -neighbourhoods.

Can any open set can be written as a union of countably many suitable -neighbourhoods? - If
*f*is a continuous function from**R**^{2}to**R**(usual metrics!) prove that the set

{ (*x*,*y*)**R**^{2}|*f*(*x*,*y*) > 0 } is an open subset of**R**^{2}.

Deduce that the open unit disc and open unit square are open sets.

Is the set { (*x*,*y*)**R**^{2}|*f*(*x*,*y*) 0 } necessarily a closed set ? - If (
*a*_{i}) is a sequence in a metric space convergent to a point , prove that is the only limit point of the set {*a*_{i}}. Give an example of a set with exactly two limit points. Give an example of a set with countably many limit points. - Let
*X*be the set {*a*,*b*,*c*,*d*,*e*}. Determine which of the following sets are topologies on*X*.

- = {
*X*, , {*a*}, {*a*,*b*}, {*a*,*c*}}

- = {
*X*, , {*a*}, {*a*,*b*}, {*a*,*c*,*d*}, {*a*,*b*,*c*,*d*}}

- = {
*X*, , {*a*}, {*a*,*b*,*c*}, {*a*,*b*,*d*}, {*a*,*b*,*c*,*d*}}

- = {
*X*, , {*a*}, {*b*}, {*a*,*b*}, {*a*,*b*,*c*}}

Solution to question 6 - = {
- Let be the set consisting of
**R**, and all intervals of the form (*q*, ) with*q***Q**. Show that is closed under all finite unions and intersections, but is not a topology on**R**. - Let be the set consisting of
**N**, and all subsets of**N**of the form {*n*,*n*+1,*n*+2, ...} for*n***N**. Prove that is a topology on**N**. What are the closed subsets of**N**? - Let be the set of all subsets of
**R**whose complements are countable, together with the empty set. Prove that is a topology on**R**. (This is called the*co-countable topology*.)

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