A CW complex $X$ is a Hausdorff space and can be interpreted as a colimit $X = colim_k X_k$ of cells $X_k $ which satisfy can inclusion relations $X_{k-1} \subset X_k$ for each $k$ and are defined inductively as pushouts in diagramm
Here for a fixed $k$ the attatching map $\overline{\phi_i}: D^{k+1} \to X_{k+1}$ has the property $\overline{\phi_i} \vert _{S^k} = \phi_i$ where $\phi_i$ determines how the boundary $\partial D^{k+1} = S^k$ is attatched.
My aim is to show that the restriction $\overline{\phi_i} \vert _{\mathring{D}^{k+1}}: \mathring{D}^{k+1} \to X_{k+1}$ is a homeomorphism.
My attempts: What I know is that $D^{k+1}$ is kompact and $X_{k+1} \subset X$ Hausdorff. Since image of a compact space is compact and compact subspaces of Hausdorff spaces are closed, we see that $ \overline{\phi_i}$ is a closed space.
Futhermore by construction of the pushout above we see that the restriction $\overline{\phi_i} \vert _{\mathring{D}^{k+1}}: \mathring{D}^{k+1} \to X_{k+1}$ is bijective. To show that it is also a homeomorphism it suffice to show that it is a closed map.
Therefore let $C \subset \mathring{D}^{k+1}$ be closed in $\mathring{D}^{k+1}$. By definition of subspace topology there exist a closed set $A \in D^{k+}$ with $C = A \cap \mathring{D}^{k+1}$. Since $\overline{\phi_i}$ is a closed map, $\overline{\phi_i}(A)$ is closed too.
My problem is to show that $\overline{\phi_i}(C)= \overline{\phi_i}(A \cap \mathring{D}^{k+1}) = \overline{\phi_i}(A) \cap X_{k+1}$ holds.
Can anybody help me to proceed with this step?