Let \(X\in \Top\), \(A\in \mathsf{Ab}\mathsf{Grp}\), \(p\in X\), and define the skyscraper sheaf by \[ \iota_p(A)(U) \da \begin{cases} A & p\in U \\ 0 & \text{else}. \end{cases} \] Show that the stalk \(\iota_p(A)_q = A\) when \(q\in \cl_X(\ts{p})\) and \(0\) otherwise, and that there is an equality of sheaves \(\iota_p(A) = \iota_*(\underline{A})\) where \(\iota: \cl_X(\ts{p}) \injects X\) is the inclusion.
Solution
Stalks. The stalk is \(\iota_p(A)_q = \colim_{U\ni q}\iota_p(A)(U)\). If \(q\in\cl_X(\ts{p})\), every open \(U\ni q\) contains \(p\), so every term of the colimit is \(A\) with identity restriction maps, and \(\iota_p(A)_q = A\). If \(q\notin\cl_X(\ts{p})\), some open \(U\ni q\) omits \(p\); then \(\iota_p(A)(U)=0\), and \(\iota_p(A)_q = 0\).
Equality with the pushforward. The sheaf \(\iota_p A \da (U\mapsto A \chi_{p\in U})\) is equal to \(\iota_* \underline{A}\):
Let \(Z\da\cl_X(\ts{p})\) with the subspace topology. An open \(U\subseteq X\) meets \(Z\) if and only if \(p\in U\). Every nonempty open subset \(V\) of \(Z\) contains \(p\), which is dense in \(Z\), so \(V\) is irreducible and hence connected. The constant sheaf \(\underline{A}(V)\da\Top(V,A)\) of locally constant functions is therefore \[ \underline{A}(V) = \begin{cases} A & V \neq \emptyset \\ 0 & V = \emptyset, \end{cases} \] with identity restriction maps between nonempty opens.
Hence \[ \iota_* \underline{A}(U) \da \underline{A} (U\cap Z) = \begin{cases} A & p\in U \\ 0 & p\notin U. \end{cases} \]
The two sheaves have the same groups on every open set and the same restriction maps, so they are equal.