Adjunction space

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological spaces, and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle A} be a subspace of ${\displaystyle Y}$. Let ${\displaystyle f:A\rightarrow X}$ be a continuous map (called the attaching map). One forms the adjunction space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X \cup_f Y} (sometimes also written as ${\displaystyle X+_{f}Y}$) by taking the disjoint union of ${\displaystyle X}$ and ${\displaystyle Y}$ and identifying ${\displaystyle a}$ with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f(a)} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a} in ${\displaystyle A}$. Formally,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X\cup_f Y = (X\sqcup Y) / \sim}

where the equivalence relation ${\displaystyle \sim }$ is generated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a\sim f(a)} for all ${\displaystyle a}$ in ${\displaystyle A}$, and the quotient is given the quotient topology. As a set, ${\displaystyle X\cup _{f}Y}$ consists of the disjoint union of ${\displaystyle X}$ and (${\displaystyle Y-A}$). The topology, however, is specified by the quotient construction.

Intuitively, one may think of ${\displaystyle Y}$ as being glued onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X} via the map ${\displaystyle f}$.