Locally Compact Topological Spaces

Let $X$ be a locally compact, Hausdorff space, and $F$ a set that has closed intersection with each compact subset of $X$. Show that $F$ is closed.

Regarding the proof of Proposition 3:

Proposition 3: Let $F$ be a compact subset of a locally compact, Hausdorff space $X$ and $O$ a neighborhood of $F$. Then there is a function $f \in C_{c}(X)$ for which $f = 1$ on $F$, $f = 0$ on $X \sim O$ and $0 \leq f \leq 1$ on $X$. The proof chooses (by the preceding proposition) a neighborhood $\mathcal{V}$ of $F$ that has compact closure contained in $O$; since $\overline{\mathcal{V}}$ is compact and Hausdorff, it is normal, and $F$ and $\overline{\mathcal{V}} \sim \mathcal{V}$ are disjoint closed subsets of $\overline{\mathcal{V}}$; Urysohn's Lemma then gives a continuous function $f$ on $\overline{\mathcal{V}}$ with $f = 1$ on $F$, $f = 0$ on $\overline{\mathcal{V}} \sim \mathcal{V}$ and $0 \leq f \leq 1$ on $\overline{\mathcal{V}}$, extended to all of $X$ by setting $f = 0$ on $X \sim \overline{\mathcal{V}}$.

Let $X$ be a locally compact, Hausdorff space and $X^{*}$ the Alexandroff one-point compactification of $X$ :

Show that the Alexandroff one-point compactification of $\mathbb {R}^{n}$ is homeomorphic to the $n$-sphere $S^{n} = \left\{ x \in \mathbb {R}^{n+1} \mid \left\| x\right\| = 1\right\}$.

Show that an open subset of a locally compact, Hausdorff space, with its subspace topology, is locally compact.

Show that a closed subset of a locally compact space, with its subspace topology, is locally compact.

Show that a locally compact, Hausdorff space $X$ is compact if and only if the set consisting of the point at infinity is an open subset of the Alexandroff one-point compactification $X^{*}$ of $X$.

Let $X$ be a locally compact, Hausdorff space. Show that the Alexandroff one-point compactification $X^{*}$ is separable if and only if $X$ is separable.

Consider the topological space $X$ consisting of the set of real numbers with the topology that has complements of countable sets as a base. Show that $X$ is not locally compact.

Provide a proof of Proposition 3 by applying Urysohn's Lemma to the Alexandroff one-point compactification of $X$.

Proposition 3: Let $K$ be a compact subset of a locally compact, Hausdorff space $X$ and $O$ a neighborhood of $K$. Then there is a function $f \in C_{c}(X)$ for which $f = 1$ on $K$, $f = 0$ on $X \sim O$ and $0 \leq f \leq 1$ on $X$.

Let $f$ continuously map the locally compact, Hausdorff space $X$ onto the topological space $Y$. Is $Y$ necessarily locally compact?

Let $X$ be a topological space and $f$ a continuous function of $X$ that has compact support. Define $K = \left\{ x \in X \mid f(x) = 1\right\}$. Show that $K$ is a compact $G_{\delta }$ set.

Let $\mathcal{O}$ be an open subset of a compact, Hausdorff space $X$. Show that the mapping of $X$ to the Alexandroff one-point compactification of $\mathcal{O}$ that is the identity on $\mathcal{O}$ and takes each point in $X \sim \mathcal{O}$ into $\omega$ is continuous.

Let $X$ and $Y$ be locally compact, Hausdorff spaces, and $f$ a continuous mapping of $X$ into $Y$. Let $X^{*}$ and $Y^{*}$ be the Alexandroff one-point compactifications of $X$ and $Y$, and $f^{*}$ the mapping of $X^{*}$ into $Y^{*}$ whose restriction to $X$ is $f$ and that takes the point at infinity in $X^{*}$ into the point at infinity in $Y^{*}$. Show that $f^{*}$ is continuous if and only if $f^{-1}(K)$ is compact whenever $K \subseteq Y$ is compact. A mapping $f$ with this property is said to be proper.

Let $X$ be a locally compact, Hausdorff space. Show that a subset $F$ of $X$ is closed if and only if $F \cap K$ is closed for each compact subset $K$ of $X$. Moreover, show that the same equivalence holds if instead of being locally compact the space $X$ is first countable.

Let $\mathcal{F}$ be a family of real-valued continuous functions on a locally compact, Hausdorff space $X$ which has the following properties:

Let $K$ be a compact $G_{\delta }$ subset of a locally compact, Hausdorff space $X$. Show that there is a decreasing sequence of continuous non-negative real-valued functions on $X$ that converges pointwise on $X$ to the characteristic function of $K$.

The Baire Category Theorem asserts that in a complete metric space the intersection of a countable collection of open dense sets is dense. At the heart of its proof lies the Cantor Intersection Theorem. Show that the Fréchet Intersection Theorem is a sufficiently strong substitute for the Cantor Intersection Theorem to provide a proof of the following assertion, by first proving it in the case in which $X$ is compact: Let $X$ be a locally compact, Hausdorff space.

Use the preceding problem to prove the following: Let $X$ be a locally compact, Hausdorff space. If $\mathcal{O}$ is an open subset of $X$ that is contained in a countable union $\bigcup_{n = 1}^{\infty } F_{n}$ of closed subsets of $X$, then the union of their interiors, $\bigcup_{n = 1}^{\infty }$ int $F_{n}$, is an open dense subset of $\mathcal{O}$.

For a map $f: X \rightarrow Y$ and a collection $\mathcal{C}$ of subsets of $Y$ we define $f^{*} \mathcal{C}$ to be the collection of subsets of $X$ given by

Show that if $\mathcal{A}$ is the $\sigma$-algebra generated by $\mathcal{C}$, then $f^{*} \mathcal{A}$ is the $\sigma$-algebra generated by $f^{*} \mathcal{C}$.

For a map $f: X \rightarrow Y$ and a collection $\mathcal{C}$ of subsets of $X$, let $\mathcal{A}$ be the $\sigma$-algebra generated by $\mathcal{C}$. If $f^{-1}[f[C]] = C$ for each $C \in \mathcal{C}$, show that $f^{-1}[f[A]] = A$ for each $A \in \mathcal{A}$.