Let $S_0$ be a smooth compact $k$-dimensional manifold with boundary and $\mathcal{E}_{\rm p}(S_0, \mathbb{D}^n)$ be the space of smooth proper embeddings into the unit disk in $\mathbb{R}^n$ with the weak topology, as defined in Hirsch. Let $M \subseteq \mathbb{D}^n$ be a smooth compact $n$-dimensional submanifold with boundary.

Let $\mathcal{C}_k(M)$ be the space of $k$-dimensional properly embedded submanifolds of $M$ with boundary, that is, $$ \mathcal{C}_k(M) = \coprod_{S} \mathcal{E}_{\rm p}(S, M)/\operatorname{Diff}(S) $$ where the union runs over each diffeomorphism type. Define $$\mathcal{E}^{M}_{\rm p}(S_0, \mathbb{D}^n) = \{\iota \colon S_0 \to \mathbb{D}^n \mid \iota \text{ is transverse to } \partial{M}\}.$$

If an embedding is transverse then $\iota({S}_0) \cap {M}$ is a submanifold of ${M}$ with boundary. Hence, there is an intersection map $\mathcal{E}^{{M}}_{\rm p}({S}_0, \mathbb{D}^n) \to \mathcal{C}_k({M})$.

Is $\mathcal{E}^{M}_{\rm p}({S}_0, \mathbb{D}^n)$ open, and is the intersection map continuous?

The first question might be answered with an argument like so: if $\iota_0$ is transverse, a nearby embedding $\iota$ will keep $\iota^{-1}(\partial M)$ close to $\iota_0^{-1}(\partial M)$, and the differential there will have not changed much, keeping transversality.

I believe the second question may be settled using an argument along the following lines. For every transverse embedding $\iota$, a basic open set $\mathcal{U}$ containing the intersection $N = \iota(S_0) \cap M$ consists of those submanifolds which are "close" to $N$. But the embeddings into $\mathbb{D}^n$ which are just as close to $\iota$ should map into $\mathcal{U}$.

However, it is not even clear how to show that a nearby embedding should have an intersection with $M$ that is even diffeomorphic to $N$, let alone close.

Let $S_0$ be a smooth compact $k$-dimensional manifold with boundary and $\mathcal{E}_{\rm p}(S_0, \mathbb{D}^n)$ be the space of smooth proper embeddings into the unit disk in $\mathbb{R}^n$ with the weak topology, as defined in Hirsch. Let $M \subseteq \operatorname{Int} \mathbb{D}^n$ be a smooth compact $n$-dimensional submanifold with boundary.

Let $\mathcal{C}_k(M)$ be the space of $k$-dimensional properly embedded submanifolds of $M$ with boundary, that is, $$ \mathcal{C}_k(M) = \coprod_{S} \mathcal{E}_{\rm p}(S, M)/\operatorname{Diff}(S) $$ where the union runs over each diffeomorphism type. Define $$\mathcal{E}^{M}_{\rm p}(S_0, \mathbb{D}^n) = \{\iota \colon S_0 \to \mathbb{D}^n \mid \iota \text{ is transverse to } \partial{M}\}.$$

If an embedding is transverse then $\iota({S}_0) \cap {M}$ is a submanifold of ${M}$ with boundary. Hence, there is an intersection map $\mathcal{E}^{{M}}_{\rm p}({S}_0, \mathbb{D}^n) \to \mathcal{C}_k({M})$.

Is $\mathcal{E}^{M}_{\rm p}({S}_0, \mathbb{D}^n)$ open, and is the intersection map continuous?

The first question might be answered with an argument like so: if $\iota_0$ is transverse, a nearby embedding $\iota$ will keep $\iota^{-1}(\partial M)$ close to $\iota_0^{-1}(\partial M)$, and the differential there will have not changed much, keeping transversality.

I believe the second question may be settled using an argument along the following lines. For every transverse embedding $\iota$, a basic open set $\mathcal{U}$ containing the intersection $N = \iota(S_0) \cap M$ consists of those submanifolds which are "close" to $N$. But the embeddings into $\mathbb{D}^n$ which are just as close to $\iota$ should map into $\mathcal{U}$.

However, it is not even clear how to show that a nearby embedding should have an intersection with $M$ that is even diffeomorphic to $N$, let alone close.

Let ${S}_0$ be a smooth compact $k$-dimensional manifold with boundary and $\mathcal{E}_{\rm p}({S}_0, \mathbb{D}^n)$ be the space of smooth proper embeddings into the unit disk in $\mathbb{R}^n$ with the weak topology, as defined in Hirsch. Let ${M} \subseteq \mathbb{D}^n$ be a smooth compact $n$-dimensional submanifold with boundary.

Let $\mathcal{C}_k({M})$ be the space of $k$-dimensional properly embedded submanifolds of ${M}$ with boundary, that is $$ \mathcal{C}_k({M}) = \coprod_{{S}} \mathcal{E}_{\rm p}({S}, {M})/\operatorname{Diff}({S}) $$ where the union runs over each diffeomorphism type. Define $$\mathcal{E}^{{M}}_{\rm p}({S}_0, \mathbb{D}^n) = \{\iota \colon {S}_0 \to \mathbb{D}^n \mid \iota \text{ is transverse to } \partial{M}\}.$$

If an embedding is transverse then $\iota({S}_0) \cap {M}$ is a submanifold of ${M}$ with boundary. Hence, there is an intersection map $\mathcal{E}^{{M}}_{\rm p}({S}_0, \mathbb{D}^n) \to \mathcal{C}_k({M})$.

Is $\mathcal{E}^{M}_{\rm p}({S}_0, \mathbb{D}^n)$ open, and is the intersection map continuous?

I believe the second question may be settled using an argument along the following lines. For every transverse embedding $\iota$, a basic open set $\mathcal{U}$ containing the intersection ${N} = \iota({S}_0) \cap {M}$ consists of those submanifolds which are "close" to ${N}$. But the embeddings into $\mathbb{D}^n$ which are just as close to $\iota$ should map into $\mathcal{U}$.

However, it is not even clear how to show that a nearby embedding should have an intersection with ${M}$ that is even diffeomorphic to ${N}$, let alone close.

Let $S_0$ be a smooth compact $k$-dimensional manifold with boundary and $\mathcal{E}_{\rm p}(S_0, \mathbb{D}^n)$ be the space of smooth proper embeddings into the unit disk in $\mathbb{R}^n$ with the weak topology, as defined in Hirsch. Let $M \subseteq \mathbb{D}^n$ be a smooth compact $n$-dimensional submanifold with boundary.

Let $\mathcal{C}_k(M)$ be the space of $k$-dimensional properly embedded submanifolds of $M$ with boundary, that is, $$ \mathcal{C}_k(M) = \coprod_{S} \mathcal{E}_{\rm p}(S, M)/\operatorname{Diff}(S) $$ where the union runs over each diffeomorphism type. Define $$\mathcal{E}^{M}_{\rm p}(S_0, \mathbb{D}^n) = \{\iota \colon S_0 \to \mathbb{D}^n \mid \iota \text{ is transverse to } \partial{M}\}.$$

If an embedding is transverse then $\iota({S}_0) \cap {M}$ is a submanifold of ${M}$ with boundary. Hence, there is an intersection map $\mathcal{E}^{{M}}_{\rm p}({S}_0, \mathbb{D}^n) \to \mathcal{C}_k({M})$.

Is $\mathcal{E}^{M}_{\rm p}({S}_0, \mathbb{D}^n)$ open, and is the intersection map continuous?

The first question might be answered with an argument like so: if $\iota_0$ is transverse, a nearby embedding $\iota$ will keep $\iota^{-1}(\partial M)$ close to $\iota_0^{-1}(\partial M)$, and the differential there will have not changed much, keeping transversality.

I believe the second question may be settled using an argument along the following lines. For every transverse embedding $\iota$, a basic open set $\mathcal{U}$ containing the intersection $N = \iota(S_0) \cap M$ consists of those submanifolds which are "close" to $N$. But the embeddings into $\mathbb{D}^n$ which are just as close to $\iota$ should map into $\mathcal{U}$.

However, it is not even clear how to show that a nearby embedding should have an intersection with $M$ that is even diffeomorphic to $N$, let alone close.