Keep in mind that just because some traffic is not flowing inside a VPN doesn't necessarily mean that it is not secure or encrypted. The easy example would be HTTPS: the information you are exchanging with the server is encrypted even if you are not on a VPN. So, no VPN doesn't necessarily equal no security.

A VPN will not make you more secure in terms of anonimity (e.g. if you are trying to perform some not-so-legal operations online), but it will make communications more secure than they would be without it. Sure, the information will be unencrypted sooner or later, but in the meantime, you are encrypting it so that anybody listening on that section of the network will not be able to access your information.

I will give you another example: suppose you are running your own VPN server at home. You are out in a coffee shop and decide to browse the internet, but you connect first to the VPN. Of course the information will not be encrypted after it has reached your VPN server, but the chances of somebody actively sniffing your home connection are much lower than somebody doing the same thing in the coffee shop.

Is a VPN really secure at the VPN providers side?
For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental class.  My question is: since $\sigma_i$ is a continuous map from the $n$-dimensional simplex to $M$, what does $a_i \sigma_i$ mean?  Is it the same kind of map?  I don't know how to make sense of this expression.
