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I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes:

A topological space $X$ is compact if every open cover has a finite subcover on $X$.

Okay, this seems to make sense. But, an example is later presented in the notes: $X$ with the trivial topology $\tau = \{X, \varnothing \}$ is compact. Again, okay, this seems to make sense, as any open cover is finite, if we look at $\tau$.

My question is: Does the cover come from the set $X$ or $\tau$? Apologies if my thought process seems unclear!

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    $\begingroup$ An open cover is a collection of open sets whose union is $X$. So the elements of the collection must be from $\tau$. $\endgroup$ – Clayton May 20 '16 at 15:31
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    $\begingroup$ The elements of the set $\tau$ are the open sets of $X$. $\endgroup$ – Lee Mosher May 20 '16 at 15:38
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    $\begingroup$ Remember that $\tau \subseteq \mathscr P (X)$ so the elements of $\tau$ are subsets of $X$. $\endgroup$ – Luis Victoria May 20 '16 at 15:48
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    $\begingroup$ Be careful to distinguish between a topological space and its set of points. In your post you have used $X$ for both senses; while that is a common abuse of notation, it also runs the risk of getting yourself confused as it seems to have done. $\endgroup$ – Hurkyl May 20 '16 at 21:39
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open cover (for a given topology $\tau$): a collection of open sets $U_{\alpha}$ (open with respect to the topology $\tau$) such that $$X \subset \underset{\alpha \in A}{\bigcup} U_{\alpha}$$

open set: A set in the topology $\tau$ of the space $X$.

compact (for a given topology $\tau$): every open cover of $X$ (with respect to the topology $\tau$) has a finite subcover.

A given open cover therefore depends on both:

  • the given space $X$
  • the topology $\tau$ given for $X$

If $\nu$ is a different topology for $X$, then $(X,\tau)$ and $(X,\nu)$ are different topological spaces. Therefore, which collections of sets are open covers of $X$ and as a result whether or not $X$ is compact will be different.

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You should use elements of the topology (open sets) to cover your space.

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Also in your lecture notes should be:

Let $X$ be a set, $(X,\mathcal{T})$ a topological space. Then a collection $\mathcal{A} \subset \mathcal{P}(X)$ is called an open cover if:

(i) $\mathcal{A}$ is a cover, i.e. $\forall x \in X, \exists A \in \mathcal{A}$ such that $x \in A$.

(ii) Elements of $\mathcal{A}$ are open, i.e. $\mathcal{A} \subset \mathcal{T}$.

So the cover most definitely comes from $\mathcal{T}$.

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An open cover is a collection of open sets whose union is $X$. Thus it is a subset of $\tau$ to begin with.

Since every subset of $\tau$ is finite, then so is every open cover.

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