Simply connected = path connected + 2nd condition

Question

I read on Wikipedia:

A topology space is called simply connected if these conditions are met:

(1) it is path connected

(2) every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

I don't understand the sentence in bold in #2. If I must preserve the two endpoints of my path as I transform it I wouldn't be able to convert it into any other path in the space that is not passing through those two original points. I am assuming that by "preserving end points" we mean we are preserving their locations on the space.

This seems limiting and would not work as far as I can tell for e.g. a sphere which is supposed to be simply connected.

What am I missing here?

Answer 1

Note that in this context, "any other such path" means "any other path between the same two points". I hope that clears things up.

Answer 2

The precise condition is this. Let $X$ be the topological space.

(1) $X$ is path connected

(2) For all $x, y \in X$ and all paths $\gamma_0, \gamma_1 : [0,1] \to X$ such that$\gamma_0(0) = \gamma_1(0) = x$ and $\gamma_0(1) = \gamma_1(1) = y$, there is a continuous map $H : [0,1] \times [0,1] \to X$ such that $$H(0,t) = \gamma_0(t)$$ $$H(1,t) = \gamma_1(t)$$ $$H(s,0) = x$$ $$H(s,1) = y$$ for all $s,t \in [0,1]$.

The map $H$ is called a homotopy from $\gamma_0$ to $\gamma_1$ relative to $\{0,1\}$. The first two equations say that $H$ is a "continuous transformation" from $\gamma_0$ to $\gamma_1$, and the latter two equations say that $H$ "preserves the basepoints throughout the transformation".

Edit: Notably, this sentence of your post is not correct:

If I must preserve the two endpoints of my path as I transform it I wouldn't be able to convert it into any other path in the space that is not passing through those two original points.

That's not what condition (2) says! Condition (2) says that if two paths have the same endpoints, then I must be able to transform one into the other while preserving the endpoints. It doesn't say anything about what should happen if two paths don't have the same endpoints, and it doesn't say that any continuous transformation from one path to the other preserves the endpoints.

Answer 3

The second sentence tell you that, if $X$ is a simply connected Topological Space then, $\forall x$, $y\in X$ and

$$\forall \alpha, \ \beta: [0,1]\rightarrow X$$

continuous such that

$\alpha(0)=\beta(0)=x$,

$\alpha(1)=\beta(1)=y$

there exists an Homotopy

$$F:[0,1]\times[0,1]\rightarrow X$$

such that

$F(t,0) = \alpha(t), \ \forall t\in [0,1]$,

$F(t,1) = \beta(t), \ \forall t\in [0,1]$.

By this, you know that there must be an Homotopy for every loops of a generic point $x\in X$ and, as the Homotopy is an equivalence relation, you know that every loop of $x$ can be contracted to a point.