How do I delete an old network connection?

I followed the setup instructions through Airport Utility and did create a secure network with a password. What bothers me is that the linksys network still appears and that my Mac has been automatically connecting to that, rather than to the secure network. I have to click "Join other networks," type in my secure network and password.

I did this and my secure network is the preferred one. When I restarted, the computer did connect to the secure network but the unscure one is still there, even though I removed it and I can still get on the net through it. What I don't want is for someone else to be able to get into my computer through this unsecure network.

I went system preferences>WiFi>Advanced and then highlighted the network and clicked the minus symbol. And that's one reason I don't understand why it still appears and seems to function.

That is the strange part. I had gone to Keychain Access yesterday and linksys was not there. I did a search of all files and there was no linksys. The linksys router is no longer attached -- it's gone. But "linksys" still appears on the WiFi drop-down and I can connect to the net through it.

I appreciate your effort and suggestions. The problem is puzzling, to say the least.

That was my first thought. However, since my Mac was automatically connecting to linksys and when I went to the ip address for linksys and entered my name and admin it recognized it. So, I continued to assume it was my router. I do hope you are correct. There doesn't seem to be any other logical explanation.

Again, I appreciate the time and effort you've devoted to this.