How to define a directed path in any space?

Question 1

I'm trying to come up with a definition of adirected path$f: I \to X$ in an arbitrary space$(X, \mathcal{T})$ which has these properties:

A directed path is apath, i.e.$f$ is continuous from the usual Euclidean subspace topology of$I = [0, 1]$ to$\mathcal{T}$.
On aHausdorff space$\mathcal{T}$, any path is directed.
On afinitely generated space$\mathcal{T}$, a directed path is a monotone map from the unit interval (with its usual order) to thespecialization preorder$\preceq_{\mathcal{T}}$ of$\mathcal{T}$, i.e. for each$t_0, t_1 \in I$,

$$t_0 \leq t_1 \implies f(t_0) \preceq_{\mathcal{T}} f(t_1).$$

My attempt

I started by considering a supposedly simpler version of the problem:

Trying to capture only loopless directed paths first. On Hausdorff spaces, the requirement is then relaxed so that any path with connected fibers (for example, an arc) is directed.
I'm assuming a$T_0$ space, to avoid having to think about equivalent elements.

My attempt at a definition of a loopless directed path is as follows:

$f$ is a path (i.e.$f$ is continuous),
$f^{-1}(\{x\})$ is an interval for each$x \in X$ (i.e. fibers are connected),
$f[[0, t]]$ is closed in$f[I]$ for each$t \in I$ ($f$ maps a prefix interval into a closed set).

I have proved that this definition has the following properties:

In a Hausdorff space, a path with connected fibers is directed. This is because any path in a Hausdorff space is a closed map.
In a locally finite space, the monotonicity condition holds. The fibers of$f$ form a finite partition of$I$ into intervals, and the values of subsequent intervals are comparable in$\preceq_{\mathcal{T}}$. The closedness condition then forces the direction.

However, I have failed to either prove or disprove the monotonicity in a finitely generated space.

The simpler problem

Does the monotonicity condition hold for loopless directed paths in all finitely generated spaces?

The more general problem

The above is my main question. However, if you happen to have an answer, then these are the more general problems (i.e. what I'm really after):

Provide a definition of a loopless directed path with the desired properties.
Provide a definition of a directed path with the desired properties. A directed path with connected fibers should be equivalent to loopless directed paths.

Question 2

The claimed closedness for monotone continuous maps into finitely generated spaces is false. Just take the constant map from $I$ to a non-closed point, say. Presumably you want $f([0,t])$ to be closed only in $f(I)$ instead.

Question 3

@DavidGao Yes and yes. Closed in $f(I)$ is the intent. Fixed.

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How to define a directed path in any space?

My attempt

The simpler problem

The more general problem

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