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Is there a good standard way of proving connectedness? Disconnectedness seems a lot easier, as you just need to find the sets. For connectedness is there a similarly straightforward approach? I think I've heard path-connectedness is often convenient, but there are some connected but not path-connected sets, so what about in those situations?

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    Generally you want to show that a set is the continuous image of a connected set, or even the continuous image of the interval. – Joel Jun 30 '14 at 21:01
  • @Joel how do you show something is a continuous image? Show that there is a continuous map? –  Jul 01 '14 at 06:46
  • The term continuous image of a connected set refers to the set you obtain after application of a continuous function at each point in the connected set. – Joel Jul 01 '14 at 13:47

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I don't know any standard way, but these equivalent definitions of a connected space help out a lot I think :

(1) The only subsets of the space, say $X$ which are both open and closed are $X$ and the empty set

(2) One cannot write $X$ as a union of two disjoint non-empty open sets

(3) There exists no surjective continuous function from $X$ to discrete space with more than one point

It also helps to know that

(a) Connectedness is a topological property of a space and that the product of two connected space is connected.

(b) If a dense subset of the space is connected then the space is also connected

Learn some standard examples of spaces that are connected, for example the sphere. Often times in topology you come across spaces that may look complicated at first but are basically constructed from these spaces.

The very fluffy Panda
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