Let us consider $(X,\tau)$ a Hausdorff topological space y let $\infty$ an element such that $\infty \notin X$. Then $(\hat X, \hat \tau)$ is topological space, where $\hat X= X \cup \lbrace \infty \rbrace$ and $\hat \tau=\tau \cup \lbrace \hat X-C :$ C is a compact subset of $(X, \tau) \rbrace$.
Well. Now I have to show that:
Be $(X_1,\tau_1)$ and $(X_2,\tau_2)$ Haussdorf spaces, locally compact and are not compact. If $(X_1,\tau_1)$ and $(X_2,\tau_2)$ are homeomorphic then $(\hat X_1,\hat \tau_1)$ and $(\hat X_2,\tau_2)$ are homeomorphic.
My attempt.
If $f$ is a homeomorphism between $X_1$ and $X_2$ then we define $\hat f : (\hat X_1,\hat \tau_1)\rightarrow (\hat X_2, \hat \tau_2)$ by $\hat f(x)=f(x),\forall x \in X_1$ and $\hat f(\infty_1)=\infty_2$. Clearly this function is bijective but I can't show that $\hat f$ is continuos and an open (or closed) map.
Any hint can be useful. Thanks