analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A topological vector space, or TVS for short, is a vector space $X$ over a topological field (usually a local field, more often than not the field of real numbers or the field of complex numbers with the usual topology) $k$ (called the ground field) equipped with a topology for which the addition and scalar multiplication maps
are continuous.
Much as a topological group is a group object in Top, so a TVS is the same as a vector space internal to $Top$ … provided that we use the two-sorted notion of vector space $(k, X)$ so that the first sort is interpreted as the topological ground field. (More generally, there is a notion of topological module which is the internalization in $Top$ of the two-sorted notion of module involving a ring sort and a module over that ring; this notion is purely equational and thus interpretable in terms of finite products. Then a TVS is a topological module whose underlying ring is a field.) N.B.: if we internalize the one-sorted notion vector space, in which the ground field is encoded in the Lawvere theory as the set of unary operations, the continuity of the unary operations gives only a continuous map ${|k|} \times X \to X$ where we use the discrete topology on $k$ – which is not what we want.
Like any topological abelian group, a TVS $X$ carries a uniform space structure generated by a basis of entourages (aka vicinities) that correspond to neighborhoods $U$ of $0$:
Thus many uniform notions (uniform continuity, completeness, etc.) carry over to the TVS context. Also from the uniformity (although it is also easy to prove directly), it follows that a TVS is completely regular, and also Hausdorff if and only if it is $T_0$ (see separation axiom). Most authors insist on the $T_0$ condition to rule out degenerate cases, but that prevents the category of TVSes from being topological over Vect. If the TVS $V$ is not Hausdorff, then the subset $V_0$ defined as the intersection of all neighborhoods of zero is a vector subspace of $V$ and the quotient vector space $V/V_0$ is Hausdorff, hence Tihonov (= completely regular Hausdorff).
The condition that scalar multiplication is continuous puts significant constraints on the topology of $X$. For example, if we assume $k$ has a base of compact absorbing neighborhoods of $0$ and $V$ is Hausdorff, then for any non-zero $v \in X$ the function
maps $k$ homeomorphically onto its image. It can then be shown that $X$ cannot (for instance) be compact (unless it is the zero space and so has no non-zero $v$); a classical theorem along these lines is that $V$ (over a local field) can be locally compact Hausdorff if and only if $V$ is finite-dimensional. (In the non-Hausdorff case, the theorems are that $X$ is compact if and only if its topology is indiscrete and that $X$ is locally compact if and only if it is a finitary direct sum of indiscrete spaces.) On the other hand, a nice property of even infinite-dimensional TVSes is that they are path-connected, or at least so in the classical cases where the ground field is $\mathbb{R}$ or $\mathbb{C}$.
More classical material should be added, particularly on locally convex spaces.
The theory of TVS can be understood as the quest to find the essence of many fundamental theorems of functional analysis of Hilbert spaces (or Banach spaces), namely to find the minimal set of assumptions that are needed for Hilbert space theorems to remain true. Examples of these are:
the Open Mapping and closed graph theorems
A central rôle in the whole theory plays duality, that is the study of locally convex spaces and their duals. A prominent example is the definition of certain concepts by duality in the theory of Schwartz distributions.
Topological vector spaces come in many flavours. The following chart provides a first overview (chart originally created and published by Greg Kuperberg on MathOverflow here, current version generated using Graphviz from lctvs dot source):
locally convex spaces: where the Hahn-Banach theorem works (assuming sufficient axioms)
bornological topological vector spaces: where bounded means continuous
The following review gives lots of important examples
See also