# MATH 560 – Introduction to Topology

Credits: 3

An introduction to the basic topological concepts. Topological spaces, metric spaces, closure, interior, and frontier operators, subspaces, separation and countability properties, bases, sub bases, convergence, continuity, homeomorphisms, compactness, connectedness, quotients and products. The course will include a brief introduction to proof techniques and set theory. Other topics in topology also may be included.

Requisites
Prerequisite: MATH 222.

Typically Offered
Spring

UGE course
No

K-State 8
None

Problem 1.

Let $p: E \rightarrow B$ be a covering map. Suppose moreover that $B$ is connected and $p^{-1}\left(b_0\right)$ has $k$-elements for a fixed $b_0 \in B$. Show that $p^{-1}(b)$ also has $k$-elements, for every $b \in B$. In this case, we say that $E$ is a $k$-fold covering of $B$ (or sometimes just a $k$ cover).

Proof .

Let $b\in B$ be any point, and let $e_1, e_2, \ldots, e_k$ be the $k$ distinct elements of $p^{-1}(b_0)$ in $E$. Since $p$ is a covering map, for each $i=1,2,\ldots,k$, there exists an open neighborhood $U_i$ of $b$ in $B$ such that $p^{-1}(U_i)$ is a disjoint union of open sets $V_{i,1}, V_{i,2},\ldots,V_{i,k}$, each homeomorphic to $U_i$ via $p$. Note that $V_{i,j}$ contains exactly one element of $p^{-1}(b)$ for each $j=1,2,\ldots,k$, since the sets $V_{i,j}$ are disjoint.

Now let $e$ be any element of $p^{-1}(b)$. Since $B$ is connected, there exists a path $\gamma : [0,1] \rightarrow B$ from $b_0$ to $b$ in $B$. By the path lifting property, there exists a unique lift $\tilde{\gamma} : [0,1] \rightarrow E$ of $\gamma$ such that $\tilde{\gamma}(0) = e_1$. Since $p^{-1}(b_0)$ has $k$ elements, there exists a permutation $\sigma$ of ${1,2,\ldots,k}$ such that $\tilde{\gamma}(1) = e_{\sigma(1)}$. Moreover, since $\tilde{\gamma}(1)$ lies in the open set $V_{\sigma(1),j}$ for some $j=1,2,\ldots,k$, we have $p(e) = p(\tilde{\gamma}(1)) = p(e_{\sigma(1)})$, which implies that $e$ also lies in the open set $V_{\sigma(1),j}$ and therefore corresponds to the unique element $e_{\sigma(1),j} \in p^{-1}(b)$ in this open set.

Thus, we have shown that every element of $p^{-1}(b)$ corresponds to a unique element of the form $e_{\sigma(1),j}$ for some permutation $\sigma$ of ${1,2,\ldots,k}$ and some $j=1,2,\ldots,k$. Therefore, $p^{-1}(b)$ also has $k$ elements. This shows that $E$ is a $k$-fold covering of $B$.

Problem 2.

Let $p: E \rightarrow B$ be a covering map.
(1) Show that if $B$ is Hausdorff, then $E$ is also.
(2) Show that if $B$ is regular, then $E$ is also.
(3) Show that if $B$ is LCH, then $E$ is also.
(4) Show that if $B$ is compact, connected, and $p^{-1}\left(b_0\right)$ is finite for some $b_0 \in B$, then $E$ is compact.

Problem 3.

1. (a) If $X$ is normal and $y$ is a point of $\beta(X)-X$, show that $y$ is not the limit of a sequence of points of $X$.
(b) Show that if $X$ is completely regular and noncompact, then $\beta(X)$ is not metrizable.