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We've defined the connectedness in topology in class in this way that a topological space is connected if the only both open and closed set is empty set or the whole set.

Now I got the explanation from Wikipedia:"Now consider the space $X$ which consists of the union of the two open intervals $(0,1)$ and $(2,3)$ of $\mathbb{R}$. The topology on $X$ is inherited as the subspace topology from the ordinary topology on the real line $\mathbb{R}$. In $X$, the set $(0,1)$ is clopen, as is the set $(2,3)$. This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen."

I understand the last line of the explanation, because it corresponds to our definition of connectedness. But Can anyone tell me why $(0,1)\bigcup(2,3)$ are both open and closed(by definition)?I understand it's open but why it's closed?

A set is said to be open if there always exists a neighborhood of each point in this set and the neighborhood also is contained in the set.

A set is said to be closed if its complement is open.

Thanks for helping me :)

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3 Answers 3

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Your topological space under consideration is $(0,1)\cup (2,3)$, therefore $(0,1)\cup (2,3)$ must be open as it is the whole set. Since complement of $(0,1)\cup (2,3)$ (relative to the space being considered) is the empty set, which is open, then $(0,1)\cup (2,3)$ is by definition closed. Basically the whole space $X$ is always both open and closed.

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Every space is both open and closed in itself. Of course, $X:=(0,1)\cup(2,3)$ is open and not closed in the real line.

Now, it's clear that both $(0,1)$ and $(2,3)$ are open in the real line, so open in $X$ in the subspace topology. Since $(0,1)=X\setminus(2,3)$, then $(0,1)$ is closed in $X$. Likewise, $(2,3)$ is closed in $X$.

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That would be my problem of understanding "open" and "closed" in topology rather than in general sense. For example all family of of sets in topology $\tau$ are open, because they are all open sets in the topology(also how they are defined defined in topology). But how do we understand exactly "closed" in topology? For example, why the empty set and the whole space are both closed and open? why the disconnected space are both closed and open? can you explain more explicitly? Thanks! –  Cancan Apr 17 '13 at 15:41
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@Cancan we define closed sets are the sets whose complement (relative to whole space) is open. –  mez Apr 17 '13 at 15:44
    
I get it! Thank you all! –  Cancan Apr 17 '13 at 15:48

The thing you may not be understanding is that whether a set is closed or open depends highly on what metric or topological space it lives inside.

Wikipedia is talking about the space $(0,1) \cup (2,3)$. $(0,1)$ is closed because it's complement is $(2,3)$ -- obviously an open set.

EDIT: it is unclear in your question whether "$(0,1) \cup (2,3)$ are both open and closed" means "$(0,1)$ and $(2,3)$ are both open and closed" or "$(0,1) \cup (2,3)$ is both open and closed".

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To get italics, do this: *italics*. –  Cameron Buie Apr 17 '13 at 15:43
    
Thanks, Cameron. –  Goos Apr 17 '13 at 15:52

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