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show that A $T_2$-space is pathwise connected iff it is arcwise connected.

This is Corollary. 31.6 in the book called general topology by Willard the author write

Proof. By the theorem 31.5 and Theorem 31.2, every path is arcwise connected. ■

Theorem 31.5 : - A Hausdorff space $X$ is a continuous image of the unit interval $I$ iff it is a Peano space.

Theorem 31.2: Every Peano space is arcwise connected.

I try to prove it in details .

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Arcwise connected always implies pathwise connected. An arc is just stronger, namely a path that is also a homeomorphism between $[0,1]$ and its image. So if we have an arc between points, we surely have a path.

So suppose $X$ is any topological Hausdorff space that is path-connected. Let $x \neq y$ in $X$ then we need to find an arc from $x$ to $y$.

We start with a path $f: [0,1] \to X$ from $X$ to $Y$. Then $P= f[[0,1]] \subseteq X$ is a Hausdorff continuous image of $[0,1]$, hence a Peano space by 31.5 in Willard.

But 31.2 says a Peano space is arcwise connected and so there is an arc $\alpha: [0,1] \to P$ from $x$ to $y$ (which are in $P$) and this is also an arc into $X$ (changing the codomain does not change the continuity and $\alpha$ is still a homomeorphism onto its image, hence an arc). So $X$ is arcwise connected.

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suppose that $X$ is $T_2$-space and pathwise connected.

There is a continuos function $f:I \to X$ such that

$f(0)=a$, $f(1)=b$ for any $a,b \in X$.

Thus, $X$ is continuos image of $I$ and $X$ is $T_2$-space

Hence, by Theorem. 31.5 $X$ is Peano space

Therefore, by Theorem. 31.2 $X$ is arcwise connected.

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  • $\begingroup$ drop the last senteces and just say $X$ is arcwise connected $\endgroup$ – Henno Brandsma Jan 6 '18 at 13:28

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