Munkres Review Chapters 1-4

(1) The ordered square is connected. Proof: Linear continua are connected.

(2) $ℝ^ω$ in the uniform topology is disconnected. Proof: The sets of bounded and unbounded sequences in $ℝ^ω$ are both open in this metric, and form a separation.

(3) The ordered square is not path connected. Proof: Suppose there exists a path from $0×0$ to $1×1$. Since $I_0^2$ is a linear continuum, this implies the path is surjective and $I_0^2$ is an image of the separable $[0,1]$. But $I_0^2$ itself is not separable, for $r×(1/4,3/4)$ for $r∈I$ is a collection of uncountably many disjoint open subsets.

(4) $ℝ_K$ is not path connected. Proof: Suppose $f : [0,1]→ℝ_K$ is a path from $0$ to $1$. Then $f^{-1}(0)$ is a closed set not containing $1$, so $r = \text{sup }f^{-1}(0) < 1$. Since $[r,1]≅[0,1]$ we may assume $f(x) > 0$ for $x > 0$.

Now, let $a_n = \text{inf } f^{-1}(1/n)$ for $n∈ℕ^+$. We see $a_{n+1} < a_n$ by connectivity of continuous images, but also $a_n \nrightarrow 0$ in $[0,1]$ since $1/n \nrightarrow 0$ in $ℝ_K$. Hence let $a > 0$ be such that $a < a_n$ for all $n$. But then $f(a) > 0$ so $1/N < f(a)$ for some $N$, implying $a_N < a$ again by connectivity of continuous images, a contradiction.

(5) The ordered square is not locally path connected. Proof: The proof of (3) extends to show that the path components of $I_0^2$ are precisely $r×[0,1]$ for $r∈[0,1]$, which are not open.

(6) $ℝ_l$ is not locally compact at any of its points. Proof: Let $x∈ℝ_l$ and $U$ be a hood of $x$. Suppose $V$ is a hood of $x$ such that $\overline{V}⊆U$ is compact. Then $x∈[a,b)⊆V$ for some $a,b∈ℝ$, and since $[a,b)$ is also closed in $\overline{V}$ this implies $[a,b)$ is compact. However, $[a,b)$ is not compact even in $ℝ$.

(7) $ℝ^ω$ in the uniform topology is not locally compact. Proof: Suppose $C$ is a compact subset of $ℝ^ω$ containing a hood of $i=(0,0,...)$. Then $B[i,ε]=[-ε,ε]^ω$ is compact for some $ε∈(0,1)$. But $\{e_n\}_{n∈ℕ^+}$ (when $e_n$ is the point with zeros in every coordinate except the $n\text{th}$ in which it is $ε$) is an infinite subset containing no limit point (as $d(e_n,e_m) = ε$ for every $n≠m$), so $[-ε,ε]^ω$ cannot even be limit point compact.

(8) $ℝ^ω$ in the uniform topology is not second countable, separable, or Lindelof. Proof: We see each pair of distinct $x,y∈\{0,1\}^ω$ are of distance $1$ in $ℝ^ω$, so that $ℝ^ω$ cannot be separable and hence not second countable. As well, $\{B(x,3/4)~|~x∈\{0,1\}^ω\}$ is an uncountable open cover of the closed subset $[0,1]^ω$, yet there are not even any proper subcovers, so $ℝ^ω$ cannot be Lindelof.

(9) $ℝ^I$ is not locally metrizable. Proof: Suppose some basis element $U=\prod_{i∈I} U_i$ of $ℝ^I$ were metrizable. Then since $U_i=ℝ$ for all but finitely many $i∈I$, and since $I-F$ for any finite subset $F⊆I$ is still uncountably infinite, we see $ℝ^I$ can be imbedded in $U$. But $ℝ^I$ itself is not metrizable, as it is not normal.

(10) $ℝ^I$ is not Lindelof. Proof: Regular Lindelof spaces are normal, which $ℝ^I$ is not.