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T0: For any two distinct points, at least one has a neighborhood not containing the other. T1: Each point has a neighborhood not containing the other for any two distinct points. T2 (Hausdorff): Each two distinct points have disjoint neighborhoods.

A limit point of a set \( S \) in a topological space is a point \( x \) such that every open set containing \( x \) intersects \( S \) in some point other than \( x \).

An accumulation point of a sequence in a topological space is a point \( x \) such that, in every neighborhood of \( x \), there are infinitely many terms of the sequence.

A function \( f: X \rightarrow Y \) between topological spaces is continuous if for every open set \( V \) in Y, the preimage \( f^{-1}(V) \) is an open set in X.

A set is closed if its complement (with respect to the entire space) is an open set.

A net is a generalization of a sequence, where the indexing set is not necessarily the natural numbers but can be any directed set. It converges to a limit if, for each neighborhood of the limit, there is some point in the indexing set after which all net elements are within the neighborhood.

A covering space for a space X is a topological space C together with a continuous surjective map \( p: C \rightarrow X \) such that for each point in X, there is some neighborhood that is evenly covered by p (preimages of the neighborhood disjointly cover C). A cover is the set of preimages of that neighborhood.

A topological space is a set X together with a collection \( \tau \) of subsets of X, called open sets, satisfying the following three axioms: (1) the empty set and X itself are in \( \tau \), (2) any union of members of \( \tau \) is also in \( \tau \), (3) any finite intersection of members of \( \tau \) is also in \( \tau \).

A topological space is path-connected if there is a continuous path joining any two points in the space, i.e., a continuous function from the interval [0,1] to the space that starts at one point and ends at the other.

A space is compact if every open cover has a finite subcover, meaning that for any collection of open sets whose union contains the space, there are finitely many of these sets that still contain the space.

A quotient topology on a set \( X \) is defined by a surjective map \( p: Y \rightarrow X \) from a topological space \( Y \), where a set in \( X \) is open if and only if its preimage under \( p \) is open in \( Y \).

A subspace \( A \) is a retract of a space \( X \) if there is a continuous map \( r: X \rightarrow A \) (the retraction) such that \( r(x) = x \) for all \( x \) in \( A \).

A homeomorphism is a bijective function \( f: X \rightarrow Y \) between two topological spaces that is continuous and whose inverse function \( f^{-1} \) is also continuous.

A topological space is connected if it cannot be divided into two disjoint nonempty open sets.

The product topology on the Cartesian product of two topological spaces \( X \times Y \) is the topology whose basis is the collection of all products of open sets, i.e., sets of the form \( U \times V \) where \( U \) is open in \( X \) and \( V \) is open in \( Y \).