Quotient Topology Examination (2.22.4,6)

James Munkres Topology, chapter 22, exercises 4,6:

4(a) Define an equivalence relation on the plane $X=ℝ^2$ given by $$x_0×y_0 \sim x_1×y_1 \text{ if } x_0+y_0^2 = x_1+y_1^2$$ Let $X^*$ be the corresponding quotient space. It is homeomorphic to a familiar space; what is it?

(b) Repeat (a) for the equivalence relation $$x_0×y_0 \sim x_1×y_1 \text{ if } x_0^2+y_0^2=x_1^2+y_1^2$$ 6. Let $ℝ_K$ denote the topology of $ℝ$ granted by the basis of the usual intervals $(a,b)$ and also the intervals $(a,b)-K$ where $K=\{1/n~|~n∈ℤ^+\}$. Let $Y$ be the quotient space induced by collapsing $K$ to a point, and let $p:ℝ_K→Y$ be the quotient map.

(a) Show $ℝ_k$ is $T_1$ non-Hausdorff.
(b) Show $p×p:ℝ_K^2→Y^2$ is not a quotient map.

Proof: (4)(a,b) The two equivalence relations are those induced by the functions $f(x,y)=x+y^2$ onto $ℝ$ and $g(x,y)=x^2+y^2$ onto $[0,∞)$, respectively. As these are both surjective, continuous maps, it suffices by Corollary 22.3(a) to prove that these functions are quotient maps. It will do to verify that they are open; as such, we observe their actions on the basis elements $$f((a,b)×(c,d))= \{ \begin{array}{lr} (a,b+d^2) & : c ≤ 0\\ (a+c^2,b+d^2) & : c > 0 \end{array}$$ To note that $g$ is open, note that in $ℝ$ and hence in $[0,∞)$ the addition of two open intervals is open, and that the squaring map on $(a,b)$ is $[0,\text{max}\{a^2,b^2\})$ if $a < 0 < b$, and $(a^2,b^2)$ if $0 < a < b$ and $(b^2,a^2)$ if $a < b < 0$.

(6)(a) We note that since each point in $ℝ_K$ constitutes a closed set, and $K$ is closed (every point not contained in $K$ has a neighborhood $(a,b)-K$ not intersecting $K$), that consequently every point in $Y$ constitutes a closed set, and thus is $T_1$. Now, assume open neighborhoods $U_0,U_K$ of $0,K∈Y$ respectively. Then $p^{-1}(U_0)$ is a neighborhood in $ℝ_K$ of $0$ not intersecting $K$, so that $p^{-1}(U_0)=(a,b)-K$ for some $a < 0 < b$. Let $n$ be such that $1/n < b$; then since $p^{-1}(U_K)$ is a neighborhood of $K$, we may assume $1/n∈(c,d)$ for some $0 < c < 1/n < d$; choose some $e∈(c,1/n)-K$ so that $f(e)∈U_0∩U_K$ and $U_0$ and $U_K$ are not disjoint.

(b) We first show that the diagonal $D=\{x×y~|~x=y\}⊆Y^2$ is not closed, particularly that $0×K$ is a limit point. To this end, assume a basis neighborhood $U_0×U_K$ of $0×K$ not intersecting $D$; this would imply disjoint neighborhoods $U_0,U_K$ about $0,K$ respectively, which is impossible by the above. However, if $S$ is the (closed) diagonal of $ℝ_K^2$, then $(p×p)^{-1}(D)=S∪(K×K)$ is closed in $ℝ_K^2$ so that $p×p$ isn't a quotient map.$~\square$