Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362712 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0475 1212272 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0476 1212272 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1148 4040208 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1161 4068112 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1168 4085824 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1168 4102832 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1176 4110568 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1176 4110568 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1643 4350528 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1657 4367928 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1662 4388856 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1746 4461912 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1747 4462168 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Proof That Every Open Cover of a Subset of a Second Countable Space is Reducible to a Second Countable Cover

Theorem

Ifis a subset of a second countable spacethen overy open cover ofis reducible to a finite cover.

Proof

Letbe a countable base forand let be an open cover ofso that

For eachexists such that

Thus

The family of setsis a subset ofhence hence it is countable sowhereis a subset ofFor eachwe can choose such that

Henceandis a countable subcover of