Topology Qual Workshop Day 2: Connectedness and Path Connectedness

Warm-up Problems:

• If X is a connected space if and only if X and the empty set are the only open and closed sets.

If X is a path connected space, then X is connected.

• (Purdue ’10) Prove or disprove: If X is path connected, and $f : X \to Y$ is continuous, then $f ( x )$ is path connected.

(1) (June $^ { \prime } 0 8 \not = \mathrm { A 2 } )$ Let $p : X \to Y$ be a quotient map.
Show that if Y is connected and moreover each set $p ^ { - 1 } ( \{ y \} )$ is connected, then X is connected.

(2) (Jan $^ { \prime } 0 8 \not = { \mathrm { B 7 } } )$ Show that if A is a proper subset of a connected space X and B is a proper subset of a connected space Y , then $( X \times Y ) \setminus ( A \times B )$ is connected.

(3) (June $^ { \prime } 0 7 \not = \mathrm { A 2 } )$ A space $( X , \tau )$ is called locally path-connected if, for every $x \in X$ every neighborhood of x contains a path-connected neighborhood of x. Show that a connected, locally path-connected space is path-connected.

(4) (Jan $^ { \prime } 0 6 \not = \mathrm { A 5 } )$ Recall that a topological space X is locally connected if for every point $x \in X$ and every neighborhood U of x there exists a connected neighborhood $V$ of x with $V \subseteq U$

(a) Prove that a topological space X is localy connected iff for every open set $U \subseteq X$ the components of U are open.

(b) Now let $p : X \to Y$ be a quotient map.
Prove that if C is a component of an open subset $U \subset Y$ then $p ^ { - 1 } ( C )$ is a union of components of $p ^ { - 1 } ( U )$ ).

(c) Deduce that if X is locally connected, then so is $Y$

(5) (June ’05 $\# \ \mathrm { A 3 } )$ Let X be a connected space with connected subset $Y .$ , and suppose that $X \setminus Y$ is not connected, with $X \setminus Y = A \cup B$ a separation of $X \setminus Y$ . Show that if B is open in X, then A  Y is a connected subset of X.

(6) (Jan $^ { \prime } 0 2 \not = \mathrm { A 3 } )$ Let X be a topological space, and A, $B \subset X$ be connected subsets of X. Show that if $A \cap { \overline { { B } } } \neq \emptyset$ then $A \cup B$ is a connected subset of X.

(7) (Jan ${ } ^ { , } 0 2 \# \mathrm { A } 5 )$ Show that R and $\mathbb { R } ^ { 2 }$ (with their usual topologies) are not homeomorphic.

(8) (June $\mathrm { { 1 0 \# A 4 } ) }$ Let $Z = X \cup Y$ , for X and Y connected subspaces of Z with $X \cap Y = \emptyset$ Let $x _ { 0 } \in X$ and $y _ { 0 } \in Y$ . Let  be the equivalence relation generated by the equivalence $x _ { 0 } \sim y _ { 0 }$ . Show that the quotient space $Z / \sim$ is connected.