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: 0.0002 362720 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0653 1212256 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0653 1212256 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1526 4050200 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1543 4078008 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1553 4095720 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1555 4126752 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1569 4149472 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1569 4149472 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2854 4779464 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2884 4880448 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2891 4901376 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3015 4969352 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.3015 4969608 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Proof That a Discrete Topological Space With Two or More Points is Disconnected

Theorem

A spaceis said to be disconnected if the setcan be expressed as the union of at least two mutually exclusive, nonempty subsets of


The setis connected if it is not disconnected.

Given a spacewith the discrete topology, takeand consider the sets

Both these sets are open andand

and (X,T) is disconnected.

Any set with the trivial topology is connected since the only nonempty open subset ofis itself.