Point Set Topology

Let $A$ and $B$ be connected subspaces of a topological space $X$, such that $A \cap \overline{B} \neq \emptyset$. Prove that $A \cup B$ is connected. If $A$ and $B$ are path connected, need $A \cup B$ be path connected?

Suppose that $A$ and $B$ are compact subspaces of spaces $X$ and $Y$ respectively, and that $N$ is an open neighborhood of

Prove that there are open sets $U \subset X$ and $V \subset Y$ such that

Let $X$ be a locally compact Hausdorff space. Let $A$ and $B$ be disjoint subsets of $X$, with $A$ compact and $B$ closed. Does there exist a continuous function $f : X \to [0,1]$ such that $f|_{A} = 0$ and $f|_{B} = 1$?

No proofs, only the correct answers to the question asked, are required for this problem.

If $X$ and $Y$ are topological spaces, the join of $X$ and $Y$ is the quotient space

where

Let $f : X \to Y$ be a continuous function from a space $X$ to a Hausdorff space $Y$. Let $C$ be a closed subspace of $Y$, and let $U$ be an open neighborhood of $f^{-1}(C)$ in $X$.

Let $X$ be a normal topological space and $A \subset X$ a closed subspace.

Let $p : E \to B$ be a covering map with $E$ locally path connected and simply connected. Let $X$ be a connected space, let $f : X \to B$ be a continuous map, and $f_{1}, f_{2} : X \to E$ be two lifts of $f$. Prove that there is a deck transformation $g : E \to E$ such that $f_{2} = g f_{1}$.

Let $p : \tilde{X} \to X$ be an $n$-sheeted covering projection, $n < \infty$. Suppose that $X$ is compact. Prove that $\tilde{X}$ is compact.

Let $\mathcal{T}$ and $\mathcal{U}$ be two different topology on $X$ such that $X$ is compact and Hausdorff with respect to both. Prove that $\mathcal{T} \not\subset \mathcal{U}$.

(Recall that $\mathcal{T} \subset \mathcal{U}$ means that every set in the topology $\mathcal{T}$ is contained in $\mathcal{U}$.)

Let $X$ be a topological space and let $A \subset X$. Show that if $C$ is a connected subset of $X$ that intersects both $A$ and $X - A$, then $C$ intersects $\mathrm{Bd}A$. (Recall that $\mathrm{Bd}A = \overline{A} \cap \overline{(X-A)}$.)

Let $(X,d)$ be a metric space. For any subspace $A \subset X$ and real number $\varepsilon > 0$, let

Suppose that $X$ is a dense subspace of a topological space $Y$. Prove or give counterexamples to the following assertions:

Let $X$ and $Y$ be topological spaces, $X = U \cup V$, and $f : X \to Y$ be a function so that $f|_{U}$ and $f|_{V}$ are continuous.

Let $q : X \to Y$ be a quotient space projection from a topological space $X$ to a connected topological space $Y$. Assume that $q^{-1}(y)$ is connected for each $y \in Y$.

Let $X$ and $Y$ be topological spaces, with $Y$ compact. Let $p : X \times Y \to X$ be the usual projection onto the first factor. Show that $p$ is a closed map.

Let $Y$ be a connected subset of the topological space, and let $Z$ be a set such that $Y$ is a subset of $Z$ and $Z$ is a subset of the closure of $Y$. Prove that $Z$ is connected.

Let $A$ be a connected subspace of a connected set $X$. If $C$ is a component of $X \setminus A$, show that $X \setminus C$ is connected.

A metric space $X$ has property $S$ if for every $\varepsilon > 0$ there is a cover of $X$ by connected sets each of which has diameter $< \varepsilon$.

Let $X$ be the topologist's sine curve defined by

Let $T = S^{1} \times S^{1}$ denote the torus.

Let $\mathcal{U} = \left\{ U_{\alpha }\right\}_{\alpha \in J}$ be an open cover of the space $X$.

Let $S$ be a set and let $F$ be a family of real valued functions on $S$ such that $f(s_{1}) = f(s_{2})$ for all $f \in F$ implies $s_{1} = s_{2}$. Prove that there exists a weakest topology in $S$ amongst all those for which all members of $F$ are continuous. Show further that the resulting topological space satisfies the Hausdorff separation axiom.

Let $K$ be a compact subset of $\mathbb {R}^{n}$ and $\left\{ B_{j}\right\}$ a sequence of open balls that covers $K$. Prove that there is a positive number $\varepsilon$ such that each $\varepsilon$-ball centered at a point of $K$ is contained in one of the ball $B_{j}$.

Let $(X_{1}, d_{1})$ and $(X_{2}, d_{2})$ be metric spaces and $f : X_{1} \to X_{2}$ a continuous surjective map such that $d_{1}(p,q) \leq d_{2}(f(p), f(q))$ for every pair of points $p, q$ in $X_{1}$.

1. If $X_{1}$ is complete, must $X_{2}$ be complete? Give a proof or a counterexample.

2. If $X_{2}$ is complete, must $X_{1}$ be complete? Give a proof or a counterexample.

Let $(M,d)$ be a nonempty complete metric space. Let $S$ map $M$ into $M$, and write $S^{2}$ for $S \circ S$, that is, $S^{2}(x) = S(S(x))$. Suppose that $S^{2}$ is a strict contraction, that is, there is a constant $\lambda < 1$ such that for all points $x, y \in M$, $d(S^{2}(x), S^{2}(y)) \leq \lambda d(x,y)$. Show that $S$ has a unique fixed point in $M$.

Let $K$ be a continuous function on the unit square $0 \leq x, y \leq 1$ satisfying $\left|K(x,y)\right| \leq 1$ for all $x$ and $y$. Show that there is a continuous function $f(x)$ on $[0,1]$ such that we have

Can there be more than one such function $f$?