- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

Example 1 in James Munkres' book, Topology (2nd Edition) reads as follows:

Munkres states that the map p is 'readily seen' to be surjective, continuous and closed.

My problem is with showing (rigorously) that it is indeed true that the map p is continuous and closed.

Regarding the continuity of a function Munkres says the following on page 102:

--------------------------------------------------------------------------

Let [itex]X[/itex] and [itex]Y[/itex] be topological spaces. A function [itex] f \ : \ X \to Y [/itex] is said to be continuous if for each open subset V of Y, the set [itex] f^{-1} (V) [/itex] is an open subset of X.

---------------------------------------------------------------------------

Yes ... fine ... but how do we use such a definition to prove or demonstrate the continuity of p in the example?

Can someone show me how we use the definition (or some theorems) in practice to demonstrate/ensure continuity?

I have a similar issue with showing p to be a closed map.

On page 137 Munkres writes the following:

--------------------------------------------------------------------------

A map [itex]p[/itex] "is said to be a closed map if for each closed set [itex]A[/itex] of [itex]X[/itex] the set [itex]p(A)[/itex] is closed in [itex]Y[/itex]"

---------------------------------------------------------------------------

Again, I understand the definition, I think, but how do we use it to indeed demonstrate/prove the closed nature of the particular map p in Munkres example?

Hope someone can help clarify the above issues?

Peter

Munkres states that the map p is 'readily seen' to be surjective, continuous and closed.

My problem is with showing (rigorously) that it is indeed true that the map p is continuous and closed.

Regarding the continuity of a function Munkres says the following on page 102:

--------------------------------------------------------------------------

Let [itex]X[/itex] and [itex]Y[/itex] be topological spaces. A function [itex] f \ : \ X \to Y [/itex] is said to be continuous if for each open subset V of Y, the set [itex] f^{-1} (V) [/itex] is an open subset of X.

---------------------------------------------------------------------------

Yes ... fine ... but how do we use such a definition to prove or demonstrate the continuity of p in the example?

Can someone show me how we use the definition (or some theorems) in practice to demonstrate/ensure continuity?

I have a similar issue with showing p to be a closed map.

On page 137 Munkres writes the following:

--------------------------------------------------------------------------

A map [itex]p[/itex] "is said to be a closed map if for each closed set [itex]A[/itex] of [itex]X[/itex] the set [itex]p(A)[/itex] is closed in [itex]Y[/itex]"

---------------------------------------------------------------------------

Again, I understand the definition, I think, but how do we use it to indeed demonstrate/prove the closed nature of the particular map p in Munkres example?

Hope someone can help clarify the above issues?

Peter