- #1

sa1988

- 222

- 23

## Homework Statement

##X = \{1,2,3\}## , ##\sigma = \big\{\emptyset , \{1,2\}, \{1,2,3\} \big\}##, topology ##\{X, \sigma\}##

##Y = \{4,5\}## , ##\tau = \big\{\emptyset , \{4\}, \{4,5\} \big\}##, topology ##\{Y, \tau\}##

##Z = \{2,3\} \subset X##

Find all the open sets in the subspace topology on ##Z## and determine all the open sets in the product topology on ##Z \times Y##

## Homework Equations

## The Attempt at a Solution

[/B]

As with previous threads, I'm hoping I'm on the right track. Just looking to check my answers.

Subspace topology on ##Z##, ##\sigma_z = \big\{\emptyset , \{2\}, \{2, 3\} \big\}##

Product topology is generated by basis ##\{ U \times V : U \in \sigma_z , V \in \tau \}##

Thus the product topology on ##Z \times Y## should be created from basis:

##\beta = \big\{\emptyset, \{(2,4)\}, \{(2,4),(2,5)\}, \{(2,4),(2,5),(3,4),(3,5)\}, \{(4,2)\}, \{(4,2),(4,3)\}, \{(4,2),(5,2)\}, \{(4,2),(4,3),(5,2),(5,3)\}\big\} ##

Thus all the open sets in the product topology on ##Z \times Y## are found in the union of all sets of basis elements in ##\beta## , plus the empty set:

## \Big\{\big( \cup U_\lambda \big)## ## \forall## ## U_\lambda \subset \beta\Big\} \cup \{\emptyset\}##

Is this all okay? Thanks.

(Many topology threads incoming - this subject is definitely my kryptonite)