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Let $X=\{0,1,2,3 \, ...\}$. Show some properties of topology $\tau = \{ \emptyset \}\cup \{ U_0, U_1, U_2, \, ... \}$ where $U_i = \{i, i+1, i+2,...\}$.

a) Show that $\tau$ is topology on $X$.

b) Is $\tau$ compact? Is $\tau$ metric?

c) Which subsets of $X$ are compact?

d) Determine which sequence of elements of $X$ converge to $1$ and which converge to $0$.

My attempt (Need to find my mistakes):

a) Surely $\emptyset, X \in \tau$. And every union of sets is in $\tau$. Also all the intersections are in $\tau$. So $\tau$ defines a topology in $X$.

b) We need to show that every cover of $X$ that has finite subcover. Cover $V$ has to include $U_0$ or otherwise $0 \not \in V$. And it doesnt matter how many of the sets $U_i \in V \, (i=1,2,...)$. These covers have finite subcover $V_0 = U_0$. This implies $\tau$ is compact.

To show that $\tau$ is metric we need to show we can define metric on it. I cant really figure out how to do it. Or if its even possible.

c) I think every subset of $X$ is compact. Because we can easily find a cover for it and a finite subcover.

d) Every sequence converges to $0$. And every sequence that doesnt contain $0$ converges to $1$?

I think there are some mistakes and I need help with the metric.

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Hints and Clarifications:

b) For compactness I figure that your book defines compactness as "any cover has a finite subcover". Well this is mostly called quasi-compactness. We say a space is compact if it is quasi-compact and Hausdorff. But that's semantics.

Anyway with your definition of compactness your proof is fine. For "is $X$ is metric or not", first show that "Every metric space is Hausdorff." Then ask yourself is $X$ Hausdorff?

c) Your proof can be made more rigorous if you notice the fact that any subset of $X$ has a smallest element (then what?)

d) Your proof is OK (why does this weird behavior happen?)

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    $\begingroup$ b) Indeed $X$ is not hausdorff so it's not metric c) Got it. d) Because it's not hausdorff. (every sequence has unique limit in hausdorff spaces) Thanks! :) $\endgroup$ – Kplusn Nov 22 '15 at 20:39

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