Transcription of Chapter V Connected Spaces
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Chapter V Connected Spaces
Chapter V. Connected Spaces 1. Introduction In this Chapter we introduce the idea of connectedness . connectedness is a topological property quite different from any property we considered in Chapters 1-4. A Connected space \ need not have any of the other topological properties we have discussed so far. Conversely, the only topological properties that imply \ is Connected are very extreme such as l\l 1 or \. has the trivial topology.. 2. connectedness Intuitively, a space is Connected if it is all in one piece; equivalently a space is disconnected if it can be written as the union of two nonempty separated pieces. To make this precise, we need to decide what separated should mean. For example, we think of as Connected even though . can be written as the union of two disjoint pieces: for example, E F where E ! and F ! . Evidently, separated should mean something more than disjoint.. On the other hand, if we remove the point ! to cut , then we probably think of the remaining space \ !
Connectedness Intuitively, a space is connected if it is all in one piece; equivalently a space is disconnected if it can be written as the union of two nonempty “separated” pieces. To make this precise, we need ... 2ÐBÑœÐBß0ÐBÑÑÞ 2 \ Clearly is a one-to-one map from onto .> a homeomorphism. ñ
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