Disjoint union and homeomorphism of canonical injection

algebraic-topologygeneral-topology

Definition:
Suppose $(X_i)_i$ is an indexed family of non-empty topological spaces. Recall: $\coprod_{i\in I}X_i = $ $\{$ $(x,i)$ $:$ $x\in X_i$ and $i\in I$ $\}$ . There is a canonical injection $\sigma_i: X_i \rightarrow \coprod_{i\in I}X_i$ , given by $\sigma_i(x)=(x,i)$. We usually identify each set $X_i$ with its image, $X_i^*=\sigma_i(X_i)$.

Problem:
Let $(X_i)$ be a collection of non-empty topological spaces.

The the canonical injection $\sigma_i:X_i \rightarrow X_i^*$ is a homeomorphism.

My attempt:

Observe that $\sigma_i$ $:$ $X_i$ $\rightarrow \coprod_{i\in I}X_i$ is by definition, continuous for each $i\in I$. Hence restricting its codomain, to $\sigma_i(X_i)$ yields a continuous function. Observe that for $i\in I$, $\sigma_i^{-1}(x,i)=x$ is the inverse of $\sigma_i$, hence $\sigma_i$ is a bijection.

My question is: How do I show that $\sigma_i^{-1}$ is also continuous?

Best Answer

Basic fact: if $f:X\to Y$ is a bijection between spaces TFAE:

  • $f$ is a homeomorphism.
  • $f$ is open and continuous.
  • $f$ is closed and continuous.

And we already proved that $\sigma_i$ is open (here) and a bijection between $ X_i$ and its image $X_i^\ast$.

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