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In the category of topological spaces, an initial object is a space which is initial in the categorical sense: it is the empty set, from which there is a unique continuous map to any other space.

The terminal object in the category of topological spaces is a one-point space: any other space has a unique continuous map to it.

The exponential object in the category of topological spaces is the function space $Y^X$, equipped with the compact-open topology, representing continuous functions from $X$ to $Y$.

In the category of topological spaces, the subobject classifier typically is the two-point space $\{0,1\}$ with a particular topology that classifies open subsets as 'true' and the complement as 'false'.

In topology, the coproduct is the disjoint union of spaces, which corresponds to the categorical notion of coproduct or sum, with the coproduct topology making each inclusion map continuous.

A pullback in topology is the categorical pullback, or fibered product, which generalizes the intersection of subspaces; it comes with a topology that makes the projections to the original spaces continuous.

In topology, a monomorphism is a continuous injection, which reflects the categorical notion of a monomorphism as an injective morphism.

The pushout in topology categorically sums spaces along a common subspace, endowing the resulting quotient space with a topology that makes the natural inclusions continuous.

An epimorphism in topology is a continuous surjection, representing the categorical concept of an epimorphism as a surjective morphism.