Regarding the hint: $p^{-1}(C)$ is closed in $E$ , so that it has no limit points outside of $p^{-1}(U)$ , but $p^{-1}(C)\cap V_{\alpha}$ is closed in $p^{-1}(U)$ . The hint immediately implies that if $B$ is a $T_{1}$ -space then so is $E$ .

(a) If $B$ is Hausdorff, take two points $x,y\in E$ , and if $p(x)=p(y)=z$ , then $z$ has a neighborhood $U$ evenly covered by $p$ so that $x$ and $y$ are in different open sets of $p^{-1}(z)$ homeomorophic to $U$ , but if $z=p(x)\neq z'=p(y)$ then $z$ and $z'$ have disjoint open neighborhoods and their preimages are disjoint open neighborhoods of $x$ and $y$ .

If $B$ is regular (locally compact Hausdorff), then using the fact that a $T_{1}$ -space (Hausdorff space) is regular (locally compact) iff for every point $x$ and its open neighborhood $U$ there is an open neighborhood $V$ of $x$ such that $\overline{V}\subset U$ (is compact), see Lemma 31.1(a) (Theorem 29.2), and knowing already that $E$ is a $T_{1}$ -space (Hausdorff space), if $y=p(x)\in p(U)\subset B$ where $p(U)$ is open ($p$ is open), we take a neighborhood $V\subset B$ of $y$ evenly covered by $p$ so that its preimage $p^{-1}(V)=\cup_{\alpha}V_{\alpha}\subset E$ , some $\alpha$ such that $x\in V_{\alpha}$ , and, finally, a neighborhood $W\subset V\cap p(U)$ such that $\overline{W}\subset V\cap p(U)$ (is compact), to have $p'=p|_{V_{\alpha}\cap U}$ , a homeomorphism between $V_{\alpha}\cap U$ and $V\cap p(U)$ , and $U'=p'^{-1}(W)$ , an open neighborhood of $x$ such that $\overline{U'}=p'^{-1}(\overline{W})\subset V_{\alpha}\cap U\subset U$ ($p'$ is a homeomorphism) is a closed (compact) subset of $V_{\alpha}\cup U$ , and of $E$ (using hint).

If $B$ is completely regular, for every point $x\in E$ and its open neighborhood $U$ , if $y=p(x)$ and its open neighborhood $V$ is evenly covered by $p$ ($p^{-1}(V)=\cup_{\alpha}V_{\alpha}$ , $x\in V_{\beta}$ ), there is $f$ such that $f(y)=1$ and $f(B-W)=f(B-V\cap p(U))=\{0\}$ (where $p(U)$ , and hence, $W$ , is open as $p$ is open), and $f':E\rightarrow[0,1]$ such that $f'(e)=f\circ p(e)$ if $e\in U\cap V_{\beta}$ and $f'(e)=0$ otherwise is continuous on both closed sets $E-\cup_{\alpha\neq\beta}V_{\alpha}$ (where it equals $f\circ p$ ) and $E-V_{\beta}$ (where it equals $0$ ), hence, on $E$ , and is as required.

(b) If $\{U_{\alpha}\}$ covers $E$ , for a point $y\in B$ choose, first, $V^{y}$ evenly covered by $p$ , $p^{-1}(V^{y})=\cup_{i=1}^{n_{y}}V_{i}^{y}$ , and then $V_{y}=\cap_{i=1}^{n_{y}}p(W_{i}^{y})$ where $W_{i}^{y}=V_{i}^{y}\cap U_{\alpha_{i}^{y}}$ for some $\alpha_{i}^{y}$ such that $W_{i}^{y}\cap p^{-1}(y)\neq\emptyset$ , noting that $V_{y}$ is then an open neighborhood of $y$ evenly covered by $p$ , then take a finite subcover of $B$ , $\{V_{y_{j}}\}_{j=1}^{n}$ , and consider $\{U_{\alpha_{k}^{y_{j}}}\}_{j\in\overline{1,n},k\in\overline{1,n_{y_{j}}}}$ such that if $x\in E$ and $p(x)\in V_{y_{j}}$ , then $x\in p^{-1}(V_{y_{j}})=\cap_{k=1}^{n_{y_{j}}}p^{-1}(p(W_{k}^{y_{j}}))\subset\cup_{i=1}^{n_{y_{j}}}U_{\alpha_{i}^{y_{j}}}$ .