## Warm-up Problems

• 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.

1. (Jan 2002, A3) Let X be a topological space, and $A , B \subseteq X$ be connected subsets of X. Show that if $A \cap { \overline { { B } } } \neq \emptyset$ , then $A \cup B$ is a connected subset of X.

2. (Jan 2002, A5) Show that R and $\mathbb { R } ^ { 2 }$ (with their usual topologies) are not homeomorphic.

3. (June 2004, A5) A topological space X is said to be contractible if the identity map ${ \dot { I } } _ { X } : X \to X$ is homotopic to a constant map $c : X \to X$ . Show that every contractible space is path-connected.

4. (June 2005, A3) Let X be a connected space with connected subset Y , and suppose that X\Y is not connected, with $X \backslash Y = A \cup B$ a separation of $X \backslash Y$ . Show that if B is open in X, then $A \cup Y$ is a connected subset of X.

5. (Jan 2006, A5) 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 locally 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 $V \subseteq Y$ then $p ^ { - 1 } ( C )$ is a union of components of $p ^ { - 1 } ( V )$

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

6. (June 2007, A2) 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.

7. (Jan 2008, B7) 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 $( X \times Y ) \backslash ( A \times B )$ is connected.

8. (June 2008, A1) 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.

9. (June 2010, A4) 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.

10. (May 2015, A2) If $X = \mathbb { R } ^ { n }$ with the usual Euclidean topology and $U \subseteq X$ is an open, connected subspace of X, show that U is also path-connected.