**Warmup question:**

Let us say that two continuous functions $f,g:[0,1]\to \mathbb R$ are ** topologically transverse** if their difference $f-g$ has only finitely many zeros, and each zero separates an interval where $f>g$ from an interval where $f<g$ (and let's say that we also impose $f(0)\not = g(0)$ and $f(1)\not = g(1)$).

Let $f_1,\ldots,f_n\in C^0([0,1])$ be continuous functions. Is it true that set $$\big\{\,g\in C^0([0,1])\,\,\big|\,\,\,\forall i,\,\,\, g\,\, \text{ is topologically transverse to } f_i\big\}$$ is dense in $C^0([0,1])$?

**What I really need:**

Let $\gamma_1,\ldots,\gamma_n$ be a finite collection of curves (Jordan Arcs) in $\mathbb R^2$. No transversality assumed amongst the $\gamma_i$. Is the set of curves that are topologically transverse to all the $\gamma_i$ dense in the $C^0$ topology?

**Even more generally:**

Let $M_1,\ldots,M_k$ be a finite collection of topological submanifolds of $\mathbb R^n$ (of various dimensions, say). Is the set of (let's say compact) topological submanifolds of $\mathbb R^n$ that are topologically transverse to all the $M_i$ dense among all submanifolds of $\mathbb R^n$, with respect to the $C^0$-topology?

(Here, I'm not exactly sure which "$C^0$-topology" on the set of all submanifolds of $\mathbb R^n$ is best adapted to my problem. The "$C^0$-distance" between two submanifolds $M,N\subset\mathbb R^n$ could be taken to mean:

(1) the Hausdorff distance (probably not what I want).

(2) $\inf_f\sup_{x\in M}|f(x)-x|$, where $f$ runs over all homeomorphism from $M$ to $N$.

(3) $\inf_f\sup_{x\in \mathbb R^n}|f(x)-x|$, where $f$ runs over all homeomorphism of $\mathbb R^n$ that map $M$ to $N$.)

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