I'm currently studying foliation theory, and to solve a problem I need the following to be true, but I can't prove nor disprove, I just have a feeling it may be false in general: if I have a smooth manifold $M,$ and a connected embedded submanifold $P$ of $M,$ which is contained in an immersed submanifold $F$ of $M,$ is $P$ also connected in the immersed submanifold topology of $F$?
At first I thought this was obviously true, but giving it a little more thinking, either I'm making some mistake or, if this is true, it is certainly not obvious. Here is what I did: $F$ may have a finer topology than its subspace topology. If $F$ was an embedded manifold, then the subspace topology of $P$ in $M$ would coincide with its the subspace topology as a subset of $F,$and then it would be connected in $F$. But with $F$ being immersed, $P$ with a topology induced by the immersed topology of $F$ may have an open set which messes with its connectedness. Any ideas, or did I made some mistake? Thanks.
This is in fact false in general. Let $M=\mathbb{R}^2$ and let $F$ be an open interval immersed in $M$ in the shape of the numeral $$\mathsf{8},$$ starting from the middle point and going around the top loop counterclockwise and then around the bottom loop clockwise. Let $P$ be a subset of the $\mathsf{8}$ which starts at the bottom left and then goes diagonally to the top right. Then $P$ is of course a connected embedded submanifold of $M$. However, $P$ is not connected in the topology of $F$, since it consists of two separate intervals at the start and end of $F$ together with one point (the middle point) which is halfway between them.