| Adjunction Space |
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| CATEGORIES ABOUT ADJUNCTION SPACE | |
| topology | |
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: Intuitively, we think of ''Y'' as being glued onto ''X'' via the map ''f''. As a set, ''X'' ∪''f'' ''Y'' consists of the disjoint union of ''X'' and (''Y'' − ''A''). The topology, however, is specified by the quotient construction. In the case where ''A'' is a Closed subspace of ''Y'' one can show that the map ''X'' → ''X'' ∪''f'' ''Y'' is a closed Embedding and (''Y'' − ''A'') → ''X'' ∪''f'' ''Y'' is an open embedding. The attaching construction is an example of a Pushout in the Category Of Topological Spaces . That is to say, the adjunction space is Universal with respect to following Commutative Diagram : Here ''i'' is the Inclusion Map and φ''X'', φ''Y'' are the maps obtained by composing the quotient map with the canonical injections into the disjoint union of ''X'' and ''Y''. One can form a more general pushout by replacing ''i'' with an arbitrary continuous map ''g'' — the construction is similar. Conversely, if ''f'' is also an inclusion the attaching construction is to simply glue ''X'' and ''Y'' together along their common subspace. Examples
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