2 Just fixed a typo and completed one question

Conlay described in $\textit{Isolated Invariant Sets and the Morse Index (1976)}$ the bases of what would be known as Conley Index Theory.

For the sake of simplicity let's think of vector fields defined on manifolds (a more general situation is enough by just considering locally compact spaces). Im going to pose a few basic notions needed for the question so even someone who is not familiar with the topic may give an answer.

A set (in the phase space) is called invariant if it is the union of solution curves. It is isolated if it is the maximal invariant set in some neighborhood of itself. A compact such neighborhood is called an isolating neighborhood for the invariant set.

An isolating neighborhood is an isolating block if the integral curves through each boundary point of the neighborhood goes immediately out of it in one or the other time direction.

And finally:

The $\textit{(Conley)}$ index is the homotopy type of the pointed space obtained from a block on collapsing the set of exit points (points in the boundary where integral curves go out the neighborhood) to one point.

Well, I've read that when we have an isolating block, say $N$, that has no exit points, we would have to collapse the empty set $\emptyset$ to one point (?), so there's a convention that the resulting space is $$(N \bigcup { \lbrace \star }, \rbrace, \star)$$ that is, the disjoint union of the space $N$ and an external point, all of it pointed at that external point).

So the questions are

• Why this convention makes sense (apart from that $\textit{it just works}$)? Is that the natural way of defining the operation "collapse to a point'' when all you have to collapse is $\textit{nothing}$ (i.e. $\emptyset$)?
• Is there a bigger frame where we don't need this convention because a more general rule contains it as a particular case?
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# Conley index for isolated invariant sets with no exit points

Conlay described in $\textit{Isolated Invariant Sets and the Morse Index (1976)}$ the bases of what would be known as Conley Index Theory.

For the sake of simplicity let's think of vector fields defined on manifolds (a more general situation is enough by just considering locally compact spaces). Im going to pose a few basic notions needed for the question so even someone who is not familiar with the topic may give an answer.

A set (in the phase space) is called invariant if it is the union of solution curves. It is isolated if it is the maximal invariant set in some neighborhood of itself. A compact such neighborhood is called an isolating neighborhood for the invariant set.

An isolating neighborhood is an isolating block if the integral curves through each boundary point of the neighborhood goes immediately out of it in one or the other time direction.

And finally:

The $\textit{(Conley)}$ index is the homotopy type of the pointed space obtained from a block on collapsing the set of exit points (points in the boundary where integral curves go out the neighborhood) to one point.

Well, I've read that when we have an isolating block, say $N$, that has no exit points, we would have to collapse the empty set $\emptyset$ to one point (?), so there's a convention that the resulting space is $$(N \bigcup { \star }, \star)$$ that is, the disjoint union of the space $N$ and an external point, all of it pointed at that external point).

So the questions are

• Why this convention makes sense (apart from that $\textit{it just works}$)?
• Is there a bigger frame where we don't need this convention because a more general rule contains it as a particular case?