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\noindent Complete three of the following five problems. In the next five problems {\bf $X$} is assumed to be a topological space. All ``maps'' given in both sections are assumed to be continuous although in a particular problem you may need to establish continuity of a particular map.
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\item A subset $A\subset X$ is said to be locally closed if for any $x\in X$ there is a neighborhood $U$ of $x$ such that $A\cap U$ is relatively closed. Prove that a locally closed set is closed.
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\item (a) \ If $X$ is infinite and has the finite complement topology show that the diagonal $\Delta \subset X \times X \, \Delta = \{(x,x) |x\in X\}$ is not closed.\\
\indent (b) \ If $X$ is an arbitrary topological space what is the necessary and sufficient condition for $\Delta$ to be closed? Prove your conjecture.
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\item $X$ is said to be locally connected if any neighborhood of any point contains a connected neighborhood. Prove that the connected components of a locally connected space are both open and closed.
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\item Suppose that $f : X\longrightarrow Y$ is continuous and for any $y\in Y, \ f^{-1}(y)$ is compact. Suppose that $\{A_j\}^\infty _{j=1}$ \ is a decreasing sequence of subsets of $X$ such that for any $j,f(A_j)=Y$ show that $f(\displaystyle \bigcap^\infty_{j=1} \, A_j)=Y$.
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\item A space $X$ is said to be completely normal if for any subsets $A$ and $B$ of $X$ such that $\bar{A} \cap B=\phi$ \ and \ $A\cap \bar{B} =\phi$, there are disjoint open sets $U_A$ \ and \ $U_B$ \ containing $A$ and $B$ respectively. Prove that any subspace of a completely normal space is normal.
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