I'm trying the following exercise from Willard's General Topology. Please verify what I did and help me at item 3.
4B Let $\Gamma$ denote the closed upper half plane $\{(x,y)|y \geq 0\}$ in $\mathbb{R}^2$. For each point in the open upper half-plane, basic nhoods will be the usual open disks (with the restriction, of course, that they be taken small enough to lie in $\Gamma$). At the points $z$ on the x-axis, the basic nhoods will be the sets $\{z\}\cup A$, where $A$ is an open disk in the upper half plane, tangent to the x-axis at $z$.
- Verify that this gives a topology on $\Gamma$.
- Compare the topology thus obtained with the usual topology on the closed upper half plane as a subset of $\mathbb{R}^2$.
- Describe the closure and interior operations in the space $\Gamma$.
This is my attemt at 1 and 2:
1) If $p \in \{(x,y)| y>0\}$, basic nhoods of $p$ are usual balls $B(y,\epsilon)$, for a small enough $\epsilon$. If $z \in \{(x,0)| x \in \mathbb{R}$, basic nhoods of $z$ are sets $\{z\}\cup A$, where $A$ is open disk tangent to x-axis. Let $B_x$ the nhood basis of $x \in \Gamma$.
I'll use theorem 4.5: Let $x \in \Gamma$.
V-a) Let $V \in B_x$. Or $V = B(x,\epsilon)$ or $V=\{x\} \cup A$, so either way $x \in V$.
V-b) Let $V_1,V_2 \in B_x$. Case 1) $x$ in x axis. Then $V_1,V_2$ are tangent to x axis, then one is contained on the other, wlg $V_1 \subseteq V_2$. Let $V_3 = V_1$, then $V_3 \in B_x$ and $V_3 \subseteq V_1 \cap V_2$. Case 2) $x$ in $\Gamma$ but not in x axis. Then $V_1 = B(x, \epsilon_1), V_2 = B(x, \epsilon_2)$, wlg $V_1 \subseteq V_2$. Let $V_3=V1$, then $V_3 \in B_x$ and $V_3 \subseteq V_1 \cap V_2$;
V-c) Let $V \in B_x$. Case 1) $x$ in x axis. $V$ is tangent to x axis. Let $V_0 = V$. If $y=x$, make $W=V$. If not, there is $B(y,\epsilon) \subseteq V$. Make $W=B(y,\epsilon)$. Case 2) $x$ in $\Gamma$ but not in x-axis. $V=B(x,\epsilon)$. Make $V=V_0$. $V$ is open, then $\forall y \in V_0 \exists B(y,\delta) \subseteq V$. Make $W=B(y,\delta)$.
Then, define $G \subseteq_{\rm{open}} \Gamma \iff \exists B \subseteq G | B \in B_x \forall x \in G$. By theorem 4.5, this defines a topology.
2) It is stronger than the usual topology, because it contains all the open sets $B(x,\epsilon)$ and those of the form $\{x\} \cup A$, which are not open in the usual.
3) This is where I need help the most. I tried defining closure and interior by this theorem 4.7:
$E^O = \{ x \in \Gamma | \text{some basic nhood of x is contained in E}\}$
$\bar{E} = \{ x \in \Gamma | \text{each basic nhood of x meets in E}\}$
I don't know how to describe these interior and closure for every set, specially for those that have points in x axis. I think those that don't have points in x axis have usual closure and interior as in usual topology. Also there is a lot of exercises like this in Willard:"describe closure and interior". What is the best approach for this kind of exercise?