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:

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

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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:

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A map \(\displaystyle p\) "is said to be a closed map if for each closed set \(\displaystyle A\) of \(\displaystyle X\) the set \(\displaystyle p(A)\) is closed in \(\displaystyle Y\)"

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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

NOTE: as an aside, if anyone can tell me how to get images to size to the size of the panel and hence be readable, I would appreciate it.

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 \(\displaystyle X\) and \(\displaystyle Y\) be topological spaces. A function \(\displaystyle f \ : \ X \to Y \) is said to be continuous if for each open subset V of Y, the set \(\displaystyle f^{-1} (V) \) 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 \(\displaystyle p\) "is said to be a closed map if for each closed set \(\displaystyle A\) of \(\displaystyle X\) the set \(\displaystyle p(A)\) is closed in \(\displaystyle Y\)"

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

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

NOTE: as an aside, if anyone can tell me how to get images to size to the size of the panel and hence be readable, I would appreciate it.

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