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Let $\tau^{*}$ the subspace euclidian topology over $[0,1]$

We define $\tau=\tau^{*} \cup${$\mathbb{R}$}.

(i) Are $\tau$ finer or coarser of the standard euclidian topology $\tau_e$?

(ii) Find the interior of $[0,+\infty)$, $[1/2,+\infty)$ and $\mathbb{Z}$

(iii)Find $\overline{[0,1]}$

(iv)Is $\mathbb{R},\tau$ an Hausdorff space?

(v) Is it compact? Connected?


(i) I think the two topologies are not comparable since $[0,1] \in \tau \setminus \tau_e$ and $(1,2) \in \tau_e \setminus \tau$.

(ii) I found $Int[0,+\infty)=[0,1]$.

$Int[1/2,+\infty)=(1/2,1]$

And $Int\mathbb{Z}=\emptyset$

(iii) $\overline{[0,1]}=\mathbb{R}$

(iv) This space is not $T_2$ because every $x \in [0,1]$ is not Hausdorff separable from the points $y \in (-\infty,0) \cup (1,+\infty)$. In fact every open neighborhood of $y$ is the whole $\mathbb{R}$, so there aren't disjoint neighborhoods for $x$ and $y$, and the space is not $T_2$.

(v) It's compact because for every open cover, {$\mathbb{R}$} is a finite open cover for $\mathbb{R}$

It's connected because there aren't open, non empty, sets such that they are a partition of $\mathbb{R}$

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Handy and obvious fact (that you already mentioned as well):

For any $x \in \mathbb{R}\setminus[0,1]$ the only open set containing $x$ is $\mathbb{R}$

This implies most of the following answers.

(i) is fine; (ii) are both correct, as no point outside $[0,1]$ be interior; (iii) is too, as all points outside $[0,1]$ are limit points of $[0,1]$; (iv) also implies that two points outside $[0,1]$ can never be separated.

As to (v): a little more argumentation would be nice:

$\mathbb{R}$ is compact, indeed, as for any open cover of $\mathbb{R}$, to cover $2$, e.g. must contain $\mathbb{R}$ as an element and then $\{\mathbb{R}\}$ is a very finite subcover.

connected: if $\mathbb{R} = U \cup V$ for open non-empty sets, then say $2 \in U$ and the fact implies that $U = \mathbb{R}$. So $U$ and $V$ are never disjoint. So $\mathbb{R}$ is connected in this topology.

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  • $\begingroup$ Thanks so much! ;) $\endgroup$ – VoB Aug 30 '17 at 7:35

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