I can close normal apps from the App manager, but the system tray ones at the top don't show up there. Example apps: Twitter, Google Play, simpleirc.
It has gingerbread android os
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Just because an application puts an icon in the "tray" when there is a notification does not mean the application is always running. Applications can listen for push notifications from their "services". These notifications get sent to your device(s) and the system passes the data to the specific application, like twitter.
In a lot of cases, the app will only handle the push notification and then go back to being idle or not running at all.
There really is no need to worry about closing applications with android. It handles the memory and idle applications very well. If there is a need for an application to have more memory, and there are apps that are idle and no longer need the memory they are "holding", android will release the memory from those applications and give it to the app that is requesting the memory.
Closing applications with a task manager gives a placebo affect. When you kill an application, you are freeing that memory for that short time, but almost instantly, android will then give that memory to something else. Android is Linux based, and on that system, free memory is wasted memory. It will always try to make sure all available memory is in use.
Killing them also ends up usually using more CPU and battery. When you kill an application, if it is needed to run because of some other background service or a push notification is received, the application will automatically start back up. But since it was killed previously, it needs to go through the entire initialization process. Which could require additional CPU and network usage.