Characteristic property of the subspace topology/Statement

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Statement

 Diagram [ilmath]\xymatrix{ Y \ar[r]^f \ar[dr]_{i_S\circ f} & S \ar@{^{(}->}[d]^{i_S}\\ & X}[/ilmath]
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath](S,\mathcal{J}_S)[/ilmath] be any subspace of [ilmath](X,\mathcal{ J })[/ilmath][Note 1]. The characteristic property of the subspace topology[1] is that:
• Given any topological space [ilmath](Y,\mathcal{ K })[/ilmath] and any map [ilmath]f:Y\rightarrow S[/ilmath] we have:
• [ilmath](f:Y\rightarrow S [/ilmath] is continuous[ilmath])\iff(i_S\circ f:Y\rightarrow X [/ilmath] is continuous[ilmath])[/ilmath]

Where [ilmath]i_S:S\rightarrow X[/ilmath] given by [ilmath]i_S:s\mapsto s[/ilmath] is the canonical injection of the subspace topology (which is itself continuous)[Note 2]

Notes

1. This means [ilmath]S\in\mathcal{P}(X)[/ilmath], or [ilmath]S\subseteq X[/ilmath] of course
2. This leads to two ways to prove the statement:
1. If we show [ilmath]i_S:S\rightarrow X[/ilmath] is continuous, then we can use the composition of continuous maps is continuous to show if [ilmath]f[/ilmath] continuous then so is [ilmath]i_S\circ f[/ilmath]
2. We can show the property the "long way" and then show [ilmath]i_S:S\rightarrow X[/ilmath] is continuous as a corollary