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: 0.0000 362456 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.1014 1211816 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.1014 1211816 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1700 4192920 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1714 4220504 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1721 4238216 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1723 4268736 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1734 4291456 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1734 4291456 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2264 4973144 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2299 5165296 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2305 5186224 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2410 5387200 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2410 5387456 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

The Basis For a Topology

A basis B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. The base 'generates' the topology T. Also

  1. The basis elements cover X.

  2. Letbe base elements and letFor eachthere is a basis elementcontainingand contained inBy induction the union of any finite number of basis elements is also a member of the basis.

If a collection B of subsets of X fails to satisfy either of these, then it is not a base for any topology on X. (It is a subbase, however, as is any collection of subsets of X.) Conversely, if B satisfies both of the conditions 1 and 2, then there is a unique topology on X for which B is a base; it is called the topology generated by B. (This topology is the intersection of all topologies on X containing B.) A sufficient but not necessary condition for B to generate a topology on X is that B is closed under intersections.

For example, the collection of all open intervals in the real line forms a basis for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty. In fact they are a base for the standard topology on the real numbers.

A base is not unique - more than one basis may generate the same topology, even if one basis is a subset of another. For example, the open intervalsis a basis for the normal topology onas is the set of open intervalsbut these two sets are completely disjoint and both properly contained in the base of all open intervals. A base need not be maximal, but a maximal base exists, given by the topology itself. In fact, any open sets in the space generated by a base may be added to the base without changing the topology.

The set S of all open intervals is a basis forbut for example, the set of all semi-infinite intervals of the formsandThen S is not a base for any topology on sinceis not a semi infinitte interval. One the other hand all semi infinite intervals of the formdoes form a basis forThe intersection of any finite number of the sets is just the smallest set, and of course2. is obviously satisfied.