42: Continuous Map

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A definition of continuous map

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of continuous map at point.
• The reader admits the local criterion for openness.

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Target Context

• The reader will have a definition of continuous map.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

For any topological spaces, $T_1$ and $T_2$, any map, $f:T_1\to T_2$, that is continuous at each point

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2: Note

The definition is in fact equivalent with this definition: for any topological spaces, $T_1$ and $T_2$, any map, $f:T_1\to T_2$, such that the preimage of any open set on $T_2$ is an open set. That is because for $\to$, for any open set, $U\subseteq T_2$ and the preimage, $S$, any point, $p\in S$, satisfies $f(p)\in U$, but $U$ is a neighborhood of $f(p)$, so, there is a neighborhood, $U_p$, of $p$ such that $f(U_p)\subseteq U$, which means that $U_p\subseteq S$, so, by the local criterion for openness, $S$ is open; for $\leftarrow$, for any neighborhood, $U$, of $f(p)$, the preimage, $S$, is an open set, which is a neighborhood of $p$ and $f(S)\subseteq U$.

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References

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