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\begin{flushright}
 A. Miller \\
 M551 \\
 Spring 2001\\
\end{flushright}

Complete the following definitions:

\prob The cofinite topology on $X$ has a basis which is

\prob The discrete topology on $X$ has a basis which is

\prob The order topology on the linear order $(L,\leq)$  has a basis which is

\prob The product topology on $X\times Y$  has a basis which is

\prob The subspace topology on $Y\subseteq X$ is

\prob The closure of the set $A\subseteq X$ is

\prob The set of limit points of the set $A\subseteq X$ is

\prob The interior of the set $A\subseteq X$ is
$$A^\circ$$

\prob The set of boundary points of the set $A\subseteq X$ is

\prob A topological space $X$ is Hausdorff iff

\prob If $f:X\to Y$ is a function and $B\subseteq Y$, then
define $f^{-1}(B)=$

\prob If $f:X\to Y$ is a function and $A\subseteq X$, then
define $f(A)=$

\prob If $f:X\to Y$ is a function, then
$f$ is one-to-one iff

\prob If $f:X\to Y$ is a function, then
$f$ is onto iff

\prob A function $f:X\to Y$ where $X$ and $Y$ are topological spaces
is continuous iff

\prob A function $f:X\to Y$ where $X$ and $Y$ are topological spaces
is a homeomorphism iff


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  Answers
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\prob The cofinite topology on $X$ has a basis
$${\cal B}=\{A\subseteq X\;\;:\;\; X\setminus A \mbox{ is finite} \}$$
A natural subbasis is the set of compliments of singletons.

\prob The discrete topology on $X$ has a basis
$${\cal B}=\{\{x\}\;\;:\;\; x\in X \}$$

\prob The order topology on the linear order $(L,\leq)$  has a basis
$${\cal B}=\{\{x\in L\;\;:\;\; a<x<b\}\;\;:\;\;
a,b\in L\cup\{-\infty,\infty\} \}$$

\prob The product topology on $X\times Y$  has a basis
$${\cal B}={\cal B}_X\times {\cal B}_Y$$
where ${\cal B}_X, {\cal B}_Y$ are bases for $X,Y$ respectively.

\prob The subspace topology on $Y\subseteq X$ is
$$\tau_Y=\{Y\cap U\;\;:\;\; U\in \tau_X\}$$
where $\tau_X$ is the topology on $X$.

\prob The closure of the set $A\subseteq X$ is
$$\overline{A}=\bigcap\{C^{\rm closed}\subseteq X\;:\; A\subseteq C\}$$

\prob The set of limit points of the set $A\subseteq X$ is $A^\prime$
where $x\in A^\prime$ iff for all $U$ an open neighborhood
of $x$ there exists $y\in U\cap A$ with $y\not=x$.

\prob The interior of the set $A\subseteq X$ is
$$A^\circ=\bigcup\{U^{\rm open}\subseteq X\;\;:\;\; U\subseteq A\}$$

\prob The set of boundary points of the set $A\subseteq X$ is
$$\partial A=\overline{A}\cap\overline{X\setminus A}$$

\prob A topological space $X$ is Hausdorff iff
any two distinct points in $X$ have  disjoint open neighborhoods.

\prob If $f:X\to Y$ is a function and $B\subseteq Y$, then
define $f^{-1}(B)=\{x\in X: f(x)\in B\}$

\prob If $f:X\to Y$ is a function and $A\subseteq X$, then
define $f(A)=\{f(a):a\in A\}$ or equivalently
$f(A)=\{b\in Y\;\;:\;\;\exists a\in A\; f(a)=b\}$

\prob If $f:X\to Y$ is a function, then
$f$ is one-to-one iff for all $x,y\in X$ if $f(x)=f(y)$, then $x=y$.

\prob If $f:X\to Y$ is a function, then
$f$ is onto iff for every $y\in Y$ there exists $x\in X$ such that $f(x)=y$.

\prob A function $f:X\to Y$ where $X$ and $Y$ are topological spaces
is continuous iff $f^{-1}(V)$ is open in $X$ for all $V$ open in $Y$.

\prob A function $f:X\to Y$ where $X$ and $Y$ are topological spaces
is a homeomorphism iff $f$ is one-to-one, onto, continuous, and open.
A function $f$ is open iff $f(U)$ is open in $Y$ for all $U$ open in $X$.
Equivalently $f$ is homeomorphism iff $f$ is a bijection and both $f$ and
$f^{-1}$ are continuous.


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